1. Points, Lines, and Angles

2. Polygons

What Is a Polygon?

A polygon is a closed, two-dimensional shape made entirely from straight line segments.

For a shape to be a polygon, it must:

• Be two-dimensional.
• Be closed, with no gaps.
• Have only straight sides.
• Have at least three sides.
• Have sides that meet at endpoints called vertices.

Triangles, rectangles, pentagons, and hexagons are all polygons.

A circle is not a polygon because it has a curved boundary.

An open figure is not a polygon because its sides do not form a closed shape.

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Parts of a Polygon

Polygons have several important features.

Sides

The straight line segments forming the boundary of a polygon are called sides.

A pentagon has:

5 sides

A hexagon has:

6 sides

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Vertices

A vertex is a point where two sides meet.

The plural of vertex is:

vertices

A polygon always has the same number of vertices as sides.

For example:

Triangle → 3 sides and 3 vertices

Pentagon → 5 sides and 5 vertices

Octagon → 8 sides and 8 vertices

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Interior Angles

An interior angle is an angle formed inside a polygon where two sides meet.

A pentagon has:

• 5 sides.
• 5 vertices.
• 5 interior angles.

The number of sides, vertices, and interior angles is always the same for a polygon.

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Naming Polygons

Polygons are usually classified according to their number of sides.

A polygon with n sides is sometimes called an:

n-gon

For example, a 15-sided polygon can be called a:

15-gon

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Triangles

A triangle is the simplest polygon.

It has:

• 3 sides.
• 3 vertices.
• 3 interior angles.

Triangles can be classified by their sides or angles.

Examples include:

• Equilateral triangles.
• Isosceles triangles.
• Scalene triangles.
• Right triangles.

The interior angles of every triangle add to:

180°

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Quadrilaterals

A quadrilateral is any polygon with four sides.

It has:

• 4 sides.
• 4 vertices.
• 4 interior angles.

Examples include:

• Squares.
• Rectangles.
• Parallelograms.
• Rhombuses.
• Trapezoids.
• Kites.

The interior angles of every quadrilateral add to:

360°

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Pentagons

A pentagon has:

• 5 sides.
• 5 vertices.
• 5 interior angles.

The prefix penta- means five.

A pentagon can be regular or irregular.

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Hexagons

A hexagon has:

• 6 sides.
• 6 vertices.
• 6 interior angles.

The prefix hexa- means six.

Hexagonal patterns appear in many places, including:

• Honeycomb structures.
• Floor tiles.
• Nuts and bolt heads.
• Crystal structures.

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Heptagons

A heptagon has:

• 7 sides.
• 7 vertices.
• 7 interior angles.

Heptagons are less common in everyday objects than triangles, quadrilaterals, and hexagons.

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Octagons

An octagon has:

• 8 sides.
• 8 vertices.
• 8 interior angles.

The prefix octa- means eight.

A familiar example is the common shape used for many stop signs.

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Regular Polygons

A regular polygon has:

• All sides equal in length.
• All interior angles equal in size.

Examples include:

• An equilateral triangle.
• A square.
• A regular pentagon.
• A regular hexagon.

For a polygon to be regular, both its sides and its angles must be equal.

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Irregular Polygons

An irregular polygon does not have all sides and angles equal.

A polygon may be irregular because:

• Its side lengths are different.
• Its angles are different.
• Both its side lengths and angles are different.

Most polygons that we encounter are not necessarily regular.

A rectangle that is not a square is an interesting example.

Its four angles are equal:

90°, 90°, 90°, 90°

but its four sides are not all equal.

Therefore, it is an irregular polygon under the definition of a regular polygon.

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Regular Triangle

A regular triangle is called an:

equilateral triangle

It has:

• 3 equal sides.
• 3 equal angles.

Each angle measures:

60°

because:

180° ÷ 3 = 60°

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Regular Quadrilateral

A regular quadrilateral is a:

square

It has:

• 4 equal sides.
• 4 equal angles.

Each angle measures:

90°

A rectangle with unequal adjacent sides is not regular because all four sides are not equal.

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Comparing a Square and Rectangle

A square and rectangle share several properties.

Both have:

• 4 sides.
• 4 vertices.
• 4 right angles.
• Opposite sides parallel.

However:

Square

All four sides are equal.

Rectangle

Opposite sides are equal, but all four sides do not have to be equal.

Therefore:

Every square is a rectangle, but not every rectangle is a square.

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Comparing a Square and Rhombus

A square and rhombus both have:

• 4 equal sides.
• Opposite sides parallel.

However, a square must have:

4 right angles

A rhombus does not necessarily have four right angles.

Therefore:

Every square is a rhombus, but not every rhombus is a square.

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Concave and Convex Polygons

Polygons can also be classified as convex or concave.

Convex Polygon

A convex polygon has no inward-pointing sections.

Every interior angle is less than:

180°

Examples include:

• Squares.
• Rectangles.
• Regular pentagons.
• Regular hexagons.

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Concave Polygon

A concave polygon has at least one section that points inward.

At least one interior angle is greater than:

180°

A useful visual test is to imagine drawing a line segment between any two points inside the polygon.

For a convex polygon, that segment remains inside the polygon.

For a concave polygon, it is possible to choose two points so that part of the segment passes outside the polygon.

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Diagonals

A diagonal is a line segment joining two non-adjacent vertices of a polygon.

For example, a quadrilateral has two diagonals.

Diagonals are useful for:

• Dividing polygons into triangles.
• Investigating symmetry.
• Calculating areas.
• Studying polygon properties.

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Interior Angles of Polygons

As the number of sides increases, the sum of the interior angles also increases.

The sum of the interior angles of a polygon is:

Interior angle sum = (n − 2) × 180°

where:

n = number of sides

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Why Does the Formula Work?

A polygon can often be divided into triangles by drawing diagonals from one vertex.

A pentagon can be divided into:

3 triangles

Each triangle contains:

180°

Therefore:

3 × 180° = 540°

So the interior angles of a pentagon add to:

540°

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Interior Angle Sum of a Quadrilateral

For a quadrilateral:

n = 4

Use:

(n − 2) × 180°

(4 − 2) × 180°

2 × 180° = 360°

Therefore:

Interior angle sum = 360°

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Interior Angle Sum of a Pentagon

For a pentagon:

n = 5

(5 − 2) × 180°

3 × 180°

= 540°

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Interior Angle Sum of a Hexagon

For a hexagon:

n = 6

(6 − 2) × 180°

4 × 180°

= 720°

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Interior Angle Sum of an Octagon

For an octagon:

n = 8

(8 − 2) × 180°

6 × 180°

= 1080°

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Interior Angles of Regular Polygons

In a regular polygon, all interior angles are equal.

Therefore:

Each interior angle = [(n − 2) × 180°] ÷ n

Example: Regular Pentagon

Interior angle sum:

540°

Number of angles:

5

Therefore:

540° ÷ 5 = 108°

Each interior angle of a regular pentagon is:

108°

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Regular Hexagon

Interior angle sum:

720°

There are six equal angles.

720° ÷ 6 = 120°

Therefore, each interior angle is:

120°

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Exterior Angles

An exterior angle is formed between one side of a polygon and the extension of an adjacent side.

For a convex polygon, one exterior angle at each vertex adds to a complete turn:

360°

For a regular polygon:

Exterior angle = 360° ÷ n

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Example: Regular Octagon

A regular octagon has:

8 sides

Each exterior angle is:

360° ÷ 8

= 45°

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Interior and Exterior Angles

At a vertex of a convex polygon, an interior angle and its adjacent exterior angle form a straight line.

Therefore:

interior angle + exterior angle = 180°

For a regular hexagon:

exterior angle = 60°

Therefore:

interior angle = 180° − 60°

= 120°

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Finding the Number of Sides

Exterior angles can also help determine how many sides a regular polygon has.

Suppose each exterior angle is:

40°

Use:

number of sides = 360° ÷ exterior angle

n = 360 ÷ 40

n = 9

Therefore, the polygon is a:

regular nonagon

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Finding Missing Angles

Suppose a quadrilateral has angles:

90°, 85°, 110°, and x

The angles must add to:

360°

Therefore:

90 + 85 + 110 + x = 360

285 + x = 360

x = 75°

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Another Missing-Angle Example

A pentagon has angles:

100°, 120°, 90°, 110°, and x

The interior angle sum is:

540°

Therefore:

100 + 120 + 90 + 110 + x = 540

420 + x = 540

x = 120°

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Perimeter of Polygons

The perimeter of a polygon is the total distance around its outside boundary.

For any polygon:

P = sum of all side lengths

For a regular polygon:

P = number of sides × side length

Example

A regular hexagon has sides of:

7 cm

Therefore:

P = 6 × 7

P = 42 cm

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Comparing Polygons

Polygons can be compared using several characteristics.

These include:

• Number of sides.
• Number of vertices.
• Side lengths.
• Angle sizes.
• Parallel sides.
• Perpendicular sides.
• Regular or irregular.
• Convex or concave.
• Lines of symmetry.
• Diagonals.

A polygon's name tells us its number of sides, but additional information is needed to fully describe its properties.

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Symmetry in Polygons

Some polygons have line symmetry.

A line of symmetry divides a figure into two matching halves.

Regular polygons have particularly predictable symmetry.

For a regular polygon:

number of lines of symmetry = number of sides

For example:

Equilateral triangle → 3

Square → 4

Regular pentagon → 5

Regular hexagon → 6

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Polygons in the Real World

Polygons appear throughout mathematics, nature, engineering, architecture, and design.

Examples include:

• Triangular roof supports.
• Rectangular doors and windows.
• Square floor tiles.
• Hexagonal honeycomb patterns.
• Octagonal signs.
• Polygonal building designs.
• Computer-generated graphics.
• Maps and property boundaries.

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Why Hexagons Are Useful

Regular hexagons can fit together without leaving gaps.

This is called tessellation.

Squares and equilateral triangles can also tessellate a flat surface.

This makes these polygons useful in:

• Tiling.
• Paving.
• Design.
• Engineering.
• Natural structures such as honeycomb.

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Polygons in Computer Graphics

Computer graphics frequently use polygons to construct images and three-dimensional models.

Complex surfaces can be approximated using large numbers of small polygons, often triangles.

More polygons can allow a digital model to represent curved surfaces more smoothly.

This makes polygon geometry important in:

• Video games.
• Animation.
• Engineering software.
• Computer-aided design.
• Scientific modelling.

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A Reliable Classification Strategy

When identifying a polygon, ask:

Is the figure closed?

If no, it is not a polygon.

Are all its sides straight?

If no, it is not a polygon.

How many sides does it have?

Use this to determine its name.

Are all sides and angles equal?

If yes, it is regular.

Does any part point inward?

If yes, it may be concave.

Then look for additional properties such as:

• Parallel sides.
• Perpendicular sides.
• Equal sides.
• Equal angles.
• Symmetry.

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Common Mistakes

Calling Any Closed Shape a Polygon

A polygon must be made entirely from straight sides.

A circle is closed but is not a polygon.

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Counting Sides Incorrectly

Trace around the entire boundary one side at a time.

Remember:

number of sides = number of vertices

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Confusing Regular with Symmetrical

A regular polygon must have:

all equal sides AND all equal angles

Having some symmetry does not automatically make a polygon regular.

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Assuming Every Rectangle Is Regular

A rectangle has equal angles, but its four sides are not necessarily equal.

Only a rectangle with four equal sides — a square — is a regular quadrilateral.

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Assuming Every Four-Sided Shape Is a Rectangle

Any four-sided polygon is a:

quadrilateral

A rectangle is only one type of quadrilateral.

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Confusing a Side and a Vertex

A side is a line segment.

A vertex is the point where two sides meet.

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Confusing Interior and Exterior Angles

Interior angles are measured inside the polygon.

Exterior angles are formed outside the polygon when a side is extended.

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Assuming Diagrams Are Drawn to Scale

A polygon may look regular without actually being regular.

Use the measurements and markings given in the problem.

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Check Your Understanding

1. Define a polygon.

2. List three characteristics a figure must have to be a polygon.

3. Why is a circle not a polygon?

4. What is a vertex?

5. How many vertices does a polygon with eight sides have?

6. Name a polygon with three sides.

7. Name a polygon with four sides.

8. Name a polygon with five sides.

9. Name a polygon with six sides.

10. Name a polygon with seven sides.

11. Name a polygon with eight sides.

12. Name a polygon with ten sides.

13. Explain the difference between a regular and irregular polygon.

14. Is every square a regular polygon? Explain.

15. Is every rectangle a regular polygon? Explain.

16. Explain the difference between a convex and concave polygon.

17. What is a diagonal?

18. Calculate the sum of the interior angles of a pentagon.

19. Calculate the sum of the interior angles of a hexagon.

20. Calculate the sum of the interior angles of an octagon.

21. Find each interior angle of a regular pentagon.

22. Find each interior angle of a regular hexagon.

23. Find each exterior angle of a regular octagon.

24. A regular polygon has exterior angles of 40°. How many sides does it have?

25. A quadrilateral has angles of 90°, 80°, 105°, and x. Find x.

26. A pentagon has angles of 95°, 105°, 120°, 100°, and x. Find x.

27. A regular hexagon has side length 8 cm. Find its perimeter.

28. Compare a square and a rectangle. Give two similarities and one difference.

29. Compare a square and a rhombus. Give two similarities and one difference.

30. Give three examples of polygons used in real-world objects or designs.

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Key Terms

• Polygon – closed two-dimensional figure made entirely from straight line segments.
• Side – line segment forming part of a polygon's boundary.
• Vertex – point where two sides meet.
• Vertices – plural of vertex.
• Interior angle – angle formed inside a polygon where two sides meet.
• Exterior angle – angle formed outside a polygon when one side is extended.
• Regular polygon – polygon with all sides equal and all interior angles equal.
• Irregular polygon – polygon that does not have all sides and angles equal.
• Convex polygon – polygon with all interior angles less than 180°.
• Concave polygon – polygon with at least one interior angle greater than 180°.
• Diagonal – line segment connecting two non-adjacent vertices.
• Triangle – three-sided polygon.
• Quadrilateral – four-sided polygon.
• Pentagon – five-sided polygon.
• Hexagon – six-sided polygon.
• Heptagon – seven-sided polygon.
• Octagon – eight-sided polygon.
• Nonagon – nine-sided polygon.
• Decagon – ten-sided polygon.
• Symmetry – property in which parts of a figure match in a balanced way.
• Tessellation – repeating arrangement of shapes covering a surface without gaps or overlaps.

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Key Takeaways

• A polygon is a closed, two-dimensional shape made from straight sides.
• A polygon must have at least three sides.
• The number of sides equals the number of vertices and interior angles.
• Polygons are named primarily according to their number of sides.
• A triangle has 3 sides.
• A quadrilateral has 4 sides.
• A pentagon has 5 sides.
• A hexagon has 6 sides.
• A heptagon has 7 sides.
• An octagon has 8 sides.
• A regular polygon has equal sides and equal angles.
• An irregular polygon does not have all sides and angles equal.
• A convex polygon has no interior angle greater than 180°.
• A concave polygon has at least one interior angle greater than 180°.
• A diagonal joins two non-adjacent vertices.
• Interior angle sum:

(n − 2) × 180°

• Each interior angle of a regular polygon:

[(n − 2) × 180°] ÷ n

• Exterior angles of a convex polygon add to:

360°

• Each exterior angle of a regular polygon:

360° ÷ n

• Polygons are important in geometry, architecture, engineering, design, tiling, mapping, and computer graphics.

A useful starting point for identifying any polygon is:

Count the sides → name the polygon → examine its sides and angles → classify its properties.

3. Triangles

What Is a Triangle?

A triangle is a polygon with:

• 3 straight sides.
• 3 vertices.
• 3 interior angles.

Triangles can have many different shapes and sizes, but every triangle has these basic characteristics.

Triangles can be classified in two different ways:

• By their side lengths.
• By their angle measures.

This means that one triangle can have two classifications at the same time.

For example, a triangle could be:

isosceles and acute

or:

scalene and right

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Classifying Triangles by Side Length

Triangles can be classified according to whether their sides have equal lengths.

The three main classifications are:

• Equilateral.
• Isosceles.
• Scalene.

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Equilateral Triangles

An equilateral triangle has:

3 equal sides

Because all three sides are equal, all three interior angles are also equal.

The angles in every triangle add to:

180°

Therefore, for an equilateral triangle:

180° ÷ 3 = 60°

Each interior angle is:

60°

So an equilateral triangle has:

3 equal sides

and:

3 angles of 60°

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Properties of Equilateral Triangles

An equilateral triangle:

• Has three equal sides.
• Has three equal angles.
• Has three 60° angles.
• Is always an acute triangle.
• Has three lines of symmetry.
• Is a regular polygon.

Because all its sides and angles are equal, an equilateral triangle is the only regular triangle.

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Isosceles Triangles

An isosceles triangle has at least two equal sides.

The angles opposite those equal sides are also equal.

For example, suppose a triangle has side lengths:

7 cm, 7 cm, and 10 cm

Two sides are equal.

Therefore, it is an:

isosceles triangle

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Base Angles of an Isosceles Triangle

An important property of isosceles triangles is:

Equal sides are opposite equal angles.

Suppose the top angle of an isosceles triangle is:

40°

The other two angles must be equal.

The total remaining angle is:

180° − 40° = 140°

Divide equally:

140° ÷ 2 = 70°

Therefore, the angles are:

40°, 70°, 70°

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Working Backward with Isosceles Triangles

Suppose an isosceles triangle has two equal angles measuring:

55°

Find the third angle.

55° + 55° = 110°

Therefore:

180° − 110° = 70°

The third angle is:

70°

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Scalene Triangles

A scalene triangle has:

3 different side lengths

Because no sides are equal, its three angles are also different.

For example, a triangle with sides:

5 cm, 7 cm, and 9 cm

is scalene.

A scalene triangle:

• Has no equal sides.
• Has no equal angles.
• Can be acute.
• Can be right.
• Can be obtuse.

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Comparing Triangles by Side Length

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Classifying Triangles by Angles

Triangles can also be classified according to their largest angle.

The three main classifications are:

• Acute.
• Right.
• Obtuse.

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Acute Triangles

An acute triangle has:

3 acute angles

This means every angle is:

less than 90°

For example:

50°, 60°, 70°

All three angles are less than 90°.

Therefore, the triangle is:

acute

An equilateral triangle is always acute because all its angles are:

60°

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Right Triangles

A right triangle contains exactly one:

90° angle

The other two angles must both be acute.

Since all three angles total 180°:

the two acute angles in a right triangle add to 90°

For example:

90° + 35° + 55° = 180°

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Parts of a Right Triangle

The side opposite the 90° angle is called the:

hypotenuse

The hypotenuse is always the longest side of a right triangle.

The other two sides are often called the:

legs

Right triangles are extremely important in:

• Construction.
• Engineering.
• Navigation.
• Trigonometry.
• Measurement.

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Obtuse Triangles

An obtuse triangle contains exactly one angle greater than:

90°

but less than:

180°

For example:

110°, 40°, 30°

Since one angle is:

110°

the triangle is:

obtuse

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Why a Triangle Cannot Have Two Obtuse Angles

Suppose a triangle had two angles greater than 90°.

Even if both were only:

91°

their sum would already be:

182°

But all three angles in a triangle must add to:

180°

Therefore, a triangle can have at most one obtuse angle.

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Why a Triangle Cannot Have Two Right Angles

Two right angles would add to:

90° + 90° = 180°

There would be no angle remaining for the third vertex.

Therefore, a triangle can have only one right angle.

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Triangle Angle Sum

One of the most important properties of triangles is:

This rule applies to:

• Equilateral triangles.
• Isosceles triangles.
• Scalene triangles.
• Acute triangles.
• Right triangles.
• Obtuse triangles.

It applies to every triangle.

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Finding a Missing Angle

Suppose a triangle has angles:

45°, 65°, and x

Since the angles total 180°:

45° + 65° + x = 180°

Add:

110° + x = 180°

Therefore:

x = 70°

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Another Missing-Angle Example

A triangle contains:

90°, 32°, and x

Calculate:

90° + 32° = 122°

Then:

x = 180° − 122°

x = 58°

The triangle has angles:

90°, 32°, 58°

Because it contains a 90° angle, it is a:

right triangle

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Two Ways to Classify the Same Triangle

A triangle should often be classified using both its sides and its angles.

For example:

Triangle A

Sides:

5 cm, 5 cm, 6 cm

Angles:

approximately 53°, 53°, 74°

Classification:

isosceles acute triangle

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Triangle B

Sides:

3 cm, 4 cm, 5 cm

Angles include:

90°

All sides are different.

Classification:

scalene right triangle

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Triangle C

Angles:

100°, 40°, 40°

Two angles are equal, so the opposite sides are equal.

Classification:

isosceles obtuse triangle

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Possible Triangle Combinations

Not every combination is possible.

An equilateral triangle can only be acute because every angle equals:

60°

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Equal Sides and Equal Angles

There is an important relationship between sides and angles.

Equal sides are opposite equal angles.

Similarly:

Equal angles are opposite equal sides.

This allows us to use information about sides to determine information about angles and vice versa.

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Longer Sides and Larger Angles

There is another important relationship:

The largest angle is opposite the longest side.

Similarly:

The smallest angle is opposite the shortest side.

For example, if a triangle has sides:

5 cm, 7 cm, 10 cm

the largest angle must be opposite the:

10 cm side

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Triangle Inequality

Not every set of three lengths can form a triangle.

For three lengths to form a triangle:

the sum of any two sides must be greater than the third side

Consider:

4 cm, 5 cm, 7 cm

Check:

4 + 5 > 7

4 + 7 > 5

5 + 7 > 4

All conditions are satisfied.

Therefore, these lengths can form a triangle.

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When a Triangle Cannot Form

Consider:

2 cm, 3 cm, 6 cm

Check the two shortest sides:

2 + 3 = 5

But:

5 < 6

The two shorter sides cannot reach each other to close the triangle.

Therefore:

2 cm, 3 cm, and 6 cm cannot form a triangle.

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What If the Sum Is Exactly Equal?

Consider:

3 cm, 4 cm, 7 cm

Here:

3 + 4 = 7

These lengths would form a straight line rather than a triangle.

Therefore, the sum must be:

greater than

the third side, not equal to it.

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Right Triangles and the Pythagorean Theorem

Right triangles have another special property.

For a right triangle:

where:

a and b are the shorter sides

and:

c is the hypotenuse

Example

A triangle has sides:

3 cm, 4 cm, 5 cm

Check:

3² + 4² = 9 + 16

= 25

and:

5² = 25

Therefore, the triangle is a:

right triangle

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Using Side Lengths to Classify Angles

The Pythagorean relationship can also help classify a triangle when all three side lengths are known.

Let:

c = longest side

Compare:

a² + b²

with:

c²

If:

a² + b² = c²

the triangle is:

right

If:

a² + b² > c²

the triangle is:

acute

If:

a² + b² < c²

the triangle is:

obtuse

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Example: Classifying from Side Lengths

Consider:

5 cm, 6 cm, 7 cm

Longest side:

c = 7

Calculate:

5² + 6² = 25 + 36 = 61

and:

7² = 49

Since:

61 > 49

the triangle is:

acute

Since all three sides are different, it is also:

scalene

Complete classification:

scalene acute triangle

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Example: Obtuse Triangle

Consider sides:

4 cm, 5 cm, 7 cm

Calculate:

4² + 5² = 16 + 25 = 41

and:

7² = 49

Since:

41 < 49

the triangle is:

obtuse

Since all sides are different:

scalene obtuse triangle

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Perimeter of a Triangle

The perimeter is the total distance around the triangle.

For sides:

a, b, and c

the perimeter is:

P = a + b + c

Example

A triangle has sides:

7 cm, 8 cm, and 10 cm

Therefore:

P = 7 + 8 + 10

P = 25 cm

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Area of a Triangle

The area of a triangle is calculated using its base and perpendicular height:

A = ½ × b × h

Suppose:

base = 12 cm

height = 7 cm

Then:

A = ½ × 12 × 7

A = 42 cm²

The height must be perpendicular to the chosen base.

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Real-World Triangles

Triangles are extremely common in structures because they can form strong, rigid frameworks.

Triangles are used in:

• Roof trusses.
• Bridges.
• Towers.
• Cranes.
• Bicycle frames.
• Road signs.
• Surveying.
• Architecture.

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Why Triangles Are Useful in Structures

A rectangle can change shape if its corners are pushed unless it is reinforced.

A triangle is much more resistant to changing shape when its side lengths remain fixed.

Engineers therefore use triangular frameworks to help create strong structures.

A rectangular frame can often be strengthened by adding a diagonal brace, which divides it into:

two triangles

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Solving Geometric Problems with Triangle Properties

When solving a triangle problem, first determine what information is given.

Ask:

• Are side lengths given?
• Are angles given?
• Are any sides marked equal?
• Is there a right-angle symbol?
• Is a missing angle required?
• Is a missing side required?
• Do I need to classify the triangle?

Then choose the appropriate property.

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Worked Problem: Isosceles Triangle

An isosceles triangle has one angle measuring:

100°

The other two angles are equal.

Calculate:

180° − 100° = 80°

Divide:

80° ÷ 2 = 40°

Therefore, the angles are:

100°, 40°, 40°

Classification:

isosceles obtuse

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Worked Problem: Right Isosceles Triangle

A triangle has:

• One 90° angle.
• Two equal sides.

Because it is isosceles, the remaining two angles are equal.

Their total is:

180° − 90° = 90°

Therefore:

90° ÷ 2 = 45°

The angles are:

45°, 45°, 90°

This is an:

isosceles right triangle

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Worked Problem: Missing Angle

A triangle has angles:

x, 2x, and 3x

Since:

x + 2x + 3x = 180°

Combine:

6x = 180°

Therefore:

x = 30°

The angles are:

30°, 60°, 90°

So the triangle is:

right

Since all angles are different, all sides are different.

Therefore, its full classification is:

scalene right triangle

---

A Reliable Classification Strategy

When classifying a triangle:

Look at the sides

3 equal → equilateral

2 equal → isosceles

No equal sides → scalene

Look at the angles

All less than 90° → acute

One exactly 90° → right

One greater than 90° → obtuse

Then combine the classifications when appropriate.

For example:

isosceles acute

isosceles right

scalene obtuse

---

Common Mistakes

Classifying Only by Appearance

Do not assume a triangle is isosceles simply because it looks symmetrical.

Use the measurements or markings provided.

---

Forgetting That Triangles Can Have Two Classifications

Isosceles describes the sides.

Obtuse describes the angles.

A triangle can therefore be both:

isosceles and obtuse

---

Calling a Triangle with a 90° Angle Acute

An acute triangle must have:

three angles less than 90°

If one angle is exactly 90°, it is a:

right triangle

---

Calling a 90° Angle Obtuse

An obtuse angle must be:

greater than 90°

Exactly 90° is a:

right angle

---

Forgetting the Triangle Angle Sum

Every triangle has:

180°

of interior angles.

Use this fact whenever one angle is missing.

---

Assuming Equal Sides Are Beside Equal Angles

Remember:

Equal sides are opposite equal angles.

---

Using the Wrong Side as the Hypotenuse

The hypotenuse is always:

• Opposite the 90° angle.
• The longest side of a right triangle.

---

Forgetting the Triangle Inequality

Three positive lengths do not automatically form a triangle.

The sum of any two sides must be greater than the third.

---

Forgetting to Square the Sides

When using the Pythagorean relationship, compare:

a² + b²

with:

c²

not:

a + b

with:

c

---

Check Your Understanding

1. What three basic features does every triangle have?

2. Name the three classifications of triangles based on side length.

3. Name the three classifications based on angles.

4. Describe an equilateral triangle.

5. What is the measure of each angle in an equilateral triangle?

6. Describe an isosceles triangle.

7. Describe a scalene triangle.

8. Describe an acute triangle.

9. Describe a right triangle.

10. Describe an obtuse triangle.

11. Classify a triangle with sides 6 cm, 6 cm, and 9 cm according to its sides.

12. Classify a triangle with sides 5 cm, 7 cm, and 8 cm according to its sides.

13. Classify a triangle with angles 60°, 60°, and 60°.

14. Classify a triangle with angles 90°, 35°, and 55° according to its angles.

15. Classify a triangle with angles 110°, 40°, and 30°.

16. A triangle has angles 45°, 65°, and x. Find x.

17. An isosceles triangle has a top angle of 40°. Find the other two angles.

18. An isosceles triangle has two equal angles of 65°. Find the third angle.

19. A right triangle has one acute angle of 28°. Find the other acute angle.

20. Explain why a triangle cannot contain two right angles.

21. Explain why a triangle cannot contain two obtuse angles.

22. Can an equilateral triangle be obtuse? Explain.

23. Can an isosceles triangle be right? If so, what would its angles be?

24. Can a scalene triangle be right? Give an example.

25. Which side of a triangle is opposite its largest angle?

26. Can side lengths 3 cm, 4 cm, and 8 cm form a triangle? Explain.

27. Can side lengths 5 cm, 6 cm, and 8 cm form a triangle? Explain.

28. Determine whether a triangle with sides 3 cm, 4 cm, and 5 cm is a right triangle.

29. A triangle has sides 4 cm, 5 cm, and 7 cm. Determine whether it is acute, right, or obtuse.

30. A triangle has angles x, 2x, and 3x. Find all three angles and classify the triangle.

---

Key Terms

• Triangle – polygon with three sides, three vertices, and three interior angles.
• Equilateral triangle – triangle with three equal sides and three equal angles.
• Isosceles triangle – triangle with at least two equal sides.
• Scalene triangle – triangle with three different side lengths.
• Acute triangle – triangle with three acute angles.
• Right triangle – triangle containing one 90° angle.
• Obtuse triangle – triangle containing one angle greater than 90°.
• Hypotenuse – longest side of a right triangle, opposite the 90° angle.
• Legs – the two shorter sides forming the right angle of a right triangle.
• Vertex – point where two sides meet.
• Interior angle – angle formed inside the triangle at a vertex.
• Triangle inequality – rule stating that the sum of any two side lengths must be greater than the third.
• Perimeter – total distance around a triangle.
• Base – side chosen as the reference when calculating area.
• Perpendicular height – shortest distance from the base to the opposite vertex measured at 90°.

---

Key Takeaways

• Every triangle has 3 sides, 3 vertices, and 3 interior angles.
• Triangles can be classified by both sides and angles.
• Equilateral triangles have three equal sides and three 60° angles.
• Isosceles triangles have at least two equal sides and corresponding equal angles.
• Scalene triangles have three different sides and three different angles.
• Acute triangles have three angles less than 90°.
• Right triangles contain exactly one 90° angle.
• Obtuse triangles contain exactly one angle greater than 90°.
• The interior angles of every triangle add to 180°.
• Equal sides are opposite equal angles.
• The longest side is opposite the largest angle.
• The shortest side is opposite the smallest angle.
• The hypotenuse is the longest side of a right triangle.
• Three lengths form a triangle only when the sum of any two is greater than the third.
• Equilateral triangles are always acute.
• Isosceles and scalene triangles can be acute, right, or obtuse.
• Triangle properties can be used to calculate missing sides and angles and solve practical geometric problems.

A useful classification check is:

Sides: equilateral, isosceles, or scalene

Angles: acute, right, or obtuse

4. Quadrilaterals

What Is a Quadrilateral?

A quadrilateral is a polygon with:

• 4 straight sides
• 4 vertices
• 4 interior angles

The word comes from terms meaning four and side.

There are many types of quadrilaterals. They are classified according to properties such as:

• Equal side lengths.
• Parallel sides.
• Perpendicular sides.
• Equal angles.
• Right angles.
• Diagonals.

---

Interior Angles of a Quadrilateral

The interior angles of every quadrilateral add to:

360°

This can be understood by dividing a quadrilateral into two triangles.

Each triangle has:

180°

Therefore:

180° + 180° = 360°

This rule applies to every quadrilateral, regardless of its shape.

---

Finding a Missing Angle

Suppose a quadrilateral has angles:

90°, 80°, 110°, and x

Since the total is 360°:

90° + 80° + 110° + x = 360°

Add the known angles:

280° + x = 360°

Therefore:

x = 80°

---

Parallel Sides

Parallel sides remain the same distance apart and never meet if extended.

The symbol for parallel is:

∥

For example:

AB ∥ CD

means that AB is parallel to CD.

Parallel sides are an important feature used to classify quadrilaterals.

---

Perpendicular Sides

Perpendicular sides meet at:

90°

The symbol for perpendicular is:

⊥

For example:

AB ⊥ BC

means that AB and BC meet at a right angle.

Squares and rectangles have perpendicular adjacent sides.

---

Squares

A square is a quadrilateral with:

• 4 equal sides.
• 4 right angles.
• 2 pairs of parallel sides.

A square has:

4 equal sides

and:

4 angles of 90°

Since:

4 × 90° = 360°

its interior angles satisfy the quadrilateral angle rule.

---

Properties of a Square

A square has:

• Four equal sides.
• Four equal angles.
• Four right angles.
• Two pairs of parallel sides.
• Adjacent sides that are perpendicular.
• Opposite sides that are equal.
• Two diagonals of equal length.
• Diagonals that bisect each other.
• Diagonals that meet at 90°.
• Four lines of symmetry.

A square has many properties because it belongs to several different quadrilateral categories.

---

Rectangles

A rectangle is a quadrilateral with:

• 4 right angles.
• Opposite sides equal.
• 2 pairs of parallel sides.

A rectangle does not need to have all four sides equal.

For example, a rectangle might measure:

8 cm × 5 cm

Opposite sides are equal:

8 cm and 8 cm

5 cm and 5 cm

and every angle is:

90°

---

Properties of a Rectangle

A rectangle has:

• Four right angles.
• Opposite sides equal.
• Opposite sides parallel.
• Adjacent sides perpendicular.
• Two diagonals of equal length.
• Diagonals that bisect each other.

A rectangle usually has:

2 lines of symmetry

A square is a special rectangle with four equal sides.

---

Parallelograms

A parallelogram is a quadrilateral with:

2 pairs of parallel opposite sides

Its opposite sides are also equal in length.

A parallelogram does not need to have right angles.

---

Properties of a Parallelogram

A parallelogram has:

• Two pairs of parallel sides.
• Opposite sides equal.
• Opposite angles equal.
• Adjacent angles that add to 180°.
• Diagonals that bisect each other.

For example, if one angle is:

65°

the opposite angle is:

65°

Each adjacent angle is:

180° − 65° = 115°

Therefore, its angles are:

65°, 115°, 65°, 115°

---

Rhombuses

A rhombus is a quadrilateral with:

4 equal sides

Opposite sides are parallel.

Unlike a square, a rhombus does not need to have four right angles.

---

Properties of a Rhombus

A rhombus has:

• Four equal sides.
• Two pairs of parallel sides.
• Opposite angles equal.
• Adjacent angles that add to 180°.
• Diagonals that bisect each other.
• Diagonals that meet at 90°.

A square is therefore a special type of rhombus.

---

Trapezoids

A trapezoid is a quadrilateral with at least one pair of parallel sides.

In some curricula, the term is defined as having exactly one pair of parallel sides. It is useful to check which convention your course uses.

The parallel sides of a trapezoid are often called its:

bases

The non-parallel sides are often called its:

legs

---

Isosceles Trapezoids

An isosceles trapezoid has equal non-parallel sides.

It also has pairs of equal angles.

For example, its angles might be:

70°, 70°, 110°, 110°

These add to:

360°

---

Comparing the Main Quadrilaterals

*This table uses the inclusive definition of trapezoid: at least one pair of parallel sides.

---

A Family of Quadrilaterals

An important idea in geometry is that quadrilateral categories can overlap.

A shape can belong to more than one category.

For example, a square satisfies the definition of:

• A quadrilateral.
• A parallelogram.
• A rectangle.
• A rhombus.

This creates a hierarchy of shapes.

---

Every Square Is a Rectangle

A rectangle must have:

• Four right angles.
• Two pairs of parallel sides.
• Opposite sides equal.

A square has all of these properties.

Therefore:

Every square is a rectangle.

However, a rectangle does not need four equal sides.

Therefore:

Not every rectangle is a square.

---

Every Square Is a Rhombus

A rhombus must have:

four equal sides

A square has four equal sides.

Therefore:

Every square is a rhombus.

However, a rhombus does not need four right angles.

Therefore:

Not every rhombus is a square.

---

Rectangles Are Parallelograms

A parallelogram must have:

two pairs of parallel opposite sides

Every rectangle has two pairs of parallel opposite sides.

Therefore:

Every rectangle is a parallelogram.

But not every parallelogram has right angles.

Therefore:

Not every parallelogram is a rectangle.

---

Rhombuses Are Parallelograms

Every rhombus has:

two pairs of parallel opposite sides

Therefore:

Every rhombus is a parallelogram.

But parallelograms do not need four equal sides.

Therefore:

Not every parallelogram is a rhombus.

---

The Special Position of the Square

A square combines the defining properties of both:

a rectangle

and:

a rhombus

It has:

four equal sides

and:

four right angles

This makes the square a particularly important quadrilateral.

---

Classification by Parallel Sides

Parallel sides can help identify a quadrilateral.

Two pairs of parallel sides

The shape could be:

• Parallelogram.
• Rectangle.
• Rhombus.
• Square.

More information is needed to classify it further.

One pair of parallel sides

Using the exclusive definition, the shape is a:

trapezoid

---

Classification by Equal Sides

Suppose a quadrilateral has:

four equal sides

It could be:

• A rhombus.
• A square.

Now examine its angles.

If all angles are:

90°

it is a square.

If the angles are not all 90°, it is a rhombus but not a square.

---

Classification by Right Angles

Suppose a quadrilateral has:

four right angles

It must be a:

rectangle

If it also has:

four equal sides

then it is more specifically a:

square

---

Classification by Parallel Sides and Angles

Suppose a quadrilateral has:

• Two pairs of parallel sides.
• Opposite sides equal.
• No right angles.
• Not all four sides equal.

It is a:

parallelogram

---

Using Side Markings in Diagrams

Geometry diagrams often use small marks to show equal lengths.

Sides with the same marking have equal lengths.

For example:

• One tick on two sides means those sides are equal.
• Two ticks on another pair indicate that those two sides are equal to each other.

Different markings do not mean equal lengths.

These symbols allow us to classify shapes even when no measurements are written.

---

Using Arrow Markings

Small arrow markings are often used to show parallel sides.

Sides with matching arrow markings are parallel.

For example:

one arrow → one pair of parallel sides

two matching arrow pairs → two pairs of parallel sides

Do not rely only on how parallel the sides appear in a drawing.

Use the markings provided.

---

Right-Angle Markings

A small square placed inside a corner means:

90°

If all four corners of a quadrilateral are right angles, the shape is at least a:

rectangle

If all four sides are also equal, it is a:

square

---

Diagonals

A diagonal is a line segment connecting two non-adjacent vertices.

Every quadrilateral has:

2 diagonals

Diagonals can provide useful information for classifying quadrilaterals.

---

Diagonals of a Square

The diagonals of a square:

• Are equal in length.
• Bisect each other.
• Meet at 90°.
• Bisect the vertex angles.

This combination of properties is especially useful for identifying squares.

---

Diagonals of a Rectangle

The diagonals of a rectangle:

• Are equal in length.
• Bisect each other.

They do not generally meet at 90°.

A square is the special case where they do.

---

Diagonals of a Rhombus

The diagonals of a rhombus:

• Bisect each other.
• Meet at 90°.
• Bisect opposite angles.

They are not generally equal in length.

Again, a square is the special case where they are equal.

---

Diagonals of a Parallelogram

The diagonals of a parallelogram:

• Bisect each other.

They are not necessarily:

• Equal.
• Perpendicular.

This helps distinguish a general parallelogram from rectangles and rhombuses.

---

Solving Angle Problems in Parallelograms

Suppose one angle of a parallelogram is:

72°

The opposite angle is also:

72°

An adjacent angle is:

180° − 72°

= 108°

Therefore, the four angles are:

72°, 108°, 72°, 108°

Check:

72 + 108 + 72 + 108 = 360°

---

Solving an Algebraic Angle Problem

Suppose adjacent angles of a parallelogram are:

x

and:

2x

Adjacent angles add to:

180°

Therefore:

x + 2x = 180°

3x = 180°

x = 60°

The other angle is:

2x = 120°

Therefore, the parallelogram has angles:

60°, 120°, 60°, 120°

---

Finding Missing Sides

Opposite sides of a parallelogram are equal.

Suppose a parallelogram has:

top side = 12 cm

left side = 7 cm

Then:

bottom side = 12 cm

and:

right side = 7 cm

Its perimeter is:

12 + 7 + 12 + 7

= 38 cm

---

Perimeter of a Square

A square has four equal sides.

Therefore:

P = 4s

Suppose:

s = 8 cm

Then:

P = 4 × 8

P = 32 cm

---

Perimeter of a Rectangle

For a rectangle:

P = 2l + 2w

Suppose:

l = 10 cm

w = 6 cm

Then:

P = 2(10) + 2(6)

P = 32 cm

---

Area of Squares and Rectangles

For a rectangle:

A = l × w

For a square:

A = s²

Example

A rectangle measures:

12 cm × 5 cm

Therefore:

A = 12 × 5

A = 60 cm²

---

Area of a Parallelogram

The area of a parallelogram is:

A = b × h

where:

b = base

and:

h = perpendicular height

The perpendicular height is not necessarily the length of the sloping side.

---

Area of a Trapezoid

The area of a trapezoid can be calculated using the lengths of its parallel sides and its perpendicular height.

If the parallel sides are:

a and b

and the perpendicular height is:

h

then:

A = ½(a + b)h

Example

Suppose:

a = 8 cm

b = 12 cm

h = 5 cm

Then:

A = ½(8 + 12)(5)

A = ½(20)(5)

A = 50 cm²

---

Quadrilaterals in the Real World

Quadrilaterals are extremely common in everyday objects and structures.

Examples include:

• Doors.
• Windows.
• Screens.
• Books.
• Tables.
• Floor tiles.
• Road signs.
• Building designs.
• Picture frames.
• Roof structures.

---

Quadrilaterals in Engineering and Design

Different quadrilateral properties can be useful for different purposes.

Rectangles are common because:

• They fit together without gaps.
• They provide right-angle corners.
• Their dimensions are easy to measure.

Parallelograms can appear in:

• Mechanical linkages.
• Folding mechanisms.
• Scissor mechanisms.

Trapezoidal shapes may be used in:

• Bridges.
• Roof structures.
• Containers.
• Structural designs.

---

Comparing Quadrilaterals

When comparing quadrilaterals, examine:

Sides

• Are any equal?
• Are all equal?

Parallel relationships

• Is there one pair?
• Are there two pairs?

Angles

• Are any 90°?
• Are opposite angles equal?

Diagonals

• Are they equal?
• Are they perpendicular?
• Do they bisect each other?

Using several properties together allows a shape to be classified accurately.

---

A Classification Strategy

Suppose you are given an unknown quadrilateral.

Ask: Are there two pairs of parallel sides?

If yes, it belongs to the parallelogram family.

Then ask:

Are all four angles 90°?

If yes, it is a rectangle.

Then ask:

Are all four sides equal?

If yes, it is a square.

Alternatively, if all four sides are equal but the angles are not all 90°, it is a rhombus.

If neither special condition applies, it is a general parallelogram.

---

Most Specific Classification

Sometimes a shape fits several categories.

Suppose a quadrilateral has:

• Four equal sides.
• Four right angles.
• Two pairs of parallel sides.

It is technically:

• A quadrilateral.
• A parallelogram.
• A rectangle.
• A rhombus.
• A square.

When asked for the most specific classification, the answer is:

square

The broader categories are still mathematically correct.

---

Common Mistakes

Thinking a Square Is Not a Rectangle

A rectangle is defined by having four right angles.

A square has four right angles.

Therefore:

a square is a rectangle

---

Thinking a Rectangle Must Have Two Long and Two Short Sides

A rectangle requires opposite sides to be equal and all angles to be 90°.

Its adjacent sides can also be equal.

When this happens, the rectangle is a:

square

---

Thinking a Rhombus Must Look Like a Diamond

A rhombus is defined by:

four equal sides

Its orientation does not matter.

A square therefore satisfies the definition of a rhombus.

---

Classifying Shapes by Appearance

Do not classify a quadrilateral simply because it "looks like" a particular shape.

Use its mathematical properties.

---

Assuming Lines Are Parallel

Sides that appear parallel in a sketch may not actually be parallel.

Look for:

• Arrow markings.
• Measurements.
• Stated information.

---

Assuming an Angle Is 90°

An angle that looks like a right angle is not necessarily 90°.

Look for a right-angle symbol or other given information.

---

Confusing Parallel and Perpendicular

Remember:

Parallel → never meet

Perpendicular → meet at 90°

---

Forgetting the Quadrilateral Angle Sum

Every quadrilateral has interior angles adding to:

360°

---

Confusing a Rhombus and a Parallelogram

Every rhombus is a parallelogram.

But not every parallelogram has four equal sides.

---

Check Your Understanding

1. Define a quadrilateral.

2. How many sides does every quadrilateral have?

3. How many vertices does every quadrilateral have?

4. What is the sum of the interior angles of a quadrilateral?

5. Describe the properties of a square.

6. Describe the properties of a rectangle.

7. Describe the properties of a parallelogram.

8. Describe the properties of a rhombus.

9. Describe a trapezoid.

10. What does it mean for two sides to be parallel?

11. What does it mean for two sides to be perpendicular?

12. Which quadrilaterals have two pairs of parallel sides?

13. Which quadrilateral has four equal sides and four right angles?

14. Which quadrilateral has four equal sides but does not necessarily have right angles?

15. Explain why every square is a rectangle.

16. Explain why not every rectangle is a square.

17. Explain why every square is a rhombus.

18. Explain why every rectangle is a parallelogram.

19. Explain why every rhombus is a parallelogram.

20. A parallelogram has one angle of 70°. Find the other three angles.

21. A quadrilateral has angles 90°, 85°, 105°, and x. Find x.

22. Adjacent angles of a parallelogram are x and 2x. Find both angles.

23. A parallelogram has one side of 12 cm and an adjacent side of 8 cm. Find its perimeter.

24. A square has a perimeter of 36 cm. Find its side length.

25. A rectangle measures 14 cm × 6 cm. Find its perimeter and area.

26. A parallelogram has a base of 10 cm and a perpendicular height of 7 cm. Find its area.

27. A trapezoid has parallel sides of 8 cm and 14 cm and a height of 5 cm. Find its area.

28. A quadrilateral has four equal sides and diagonals that meet at 90°. What classifications could apply?

29. A quadrilateral has four right angles and two pairs of parallel sides. What additional information would you need to determine whether it is a square?

30. Explain how you would classify an unknown quadrilateral using its sides, angles, and parallel relationships.

---

Key Terms

• Quadrilateral – polygon with four sides.
• Parallel sides – sides that remain the same distance apart and never meet.
• Perpendicular sides – sides that meet at 90°.
• Square – quadrilateral with four equal sides and four right angles.
• Rectangle – quadrilateral with four right angles.
• Parallelogram – quadrilateral with two pairs of parallel opposite sides.
• Rhombus – quadrilateral with four equal sides.
• Trapezoid – quadrilateral with at least one pair of parallel sides under the inclusive definition.
• Diagonal – line segment joining two non-adjacent vertices.
• Opposite sides – sides that do not share a vertex.
• Adjacent sides – sides sharing a vertex.
• Opposite angles – angles at opposite vertices.
• Adjacent angles – angles sharing a side.
• Bisect – divide into two equal parts.
• Right angle – angle measuring 90°.

---

Key Takeaways

• Every quadrilateral has 4 sides, 4 vertices, and 4 interior angles.
• The interior angles of every quadrilateral add to 360°.
• A square has four equal sides and four right angles.
• A rectangle has four right angles and two pairs of parallel sides.
• A parallelogram has two pairs of parallel opposite sides.
• A rhombus has four equal sides and two pairs of parallel sides.
• A trapezoid has at least one pair of parallel sides under the inclusive definition.
• Squares and rectangles have perpendicular adjacent sides.
• Squares, rectangles, rhombuses, and parallelograms all have two pairs of parallel sides.
• Opposite angles of parallelograms are equal.
• Adjacent angles of parallelograms add to 180°.
• Every square is both a rectangle and a rhombus.
• Every rectangle is a parallelogram.
• Every rhombus is a parallelogram.
• Not every rectangle is a square.
• Not every rhombus is a square.
• Quadrilateral categories can overlap because classification is based on properties rather than appearance.

The most important relationship to remember is:

Square → Rectangle and Rhombus → Parallelogram → Quadrilateral

A square does not stop being a rectangle or rhombus simply because square is its more specific name.

5. Circles

What Is a Circle?

A circle is a two-dimensional shape consisting of all points that are the same distance from a fixed point called the center.

Unlike polygons, circles:

• Have no straight sides.
• Have no vertices.
• Have no corners.
• Have a continuously curved boundary.

Circles appear throughout mathematics and in everyday objects such as wheels, clocks, coins, plates, gears, pipes, and circular tracks.

---

The Center

The center is the point exactly in the middle of a circle.

Every point on the circumference is the same distance from the center.

The center is often labelled with a capital letter such as:

O

The center is important because the radius, diameter, and many other measurements of a circle are defined using it.

---

The Radius

The radius is the distance from the center of a circle to any point on its circumference.

The symbol commonly used for radius is:

r

Because every point on a circle is the same distance from its center, all radii of the same circle have equal lengths.

For example, if a circle has:

r = 5 cm

then every radius drawn from the center to the circumference measures:

5 cm

---

The Diameter

The diameter is a straight line segment that:

• Begins on the circumference.
• Passes through the center.
• Ends on the opposite side of the circumference.

The symbol commonly used for diameter is:

d

The diameter is always twice as long as the radius.

Therefore:

d = 2r

---

Radius and Diameter

The relationship between radius and diameter is one of the most important circle relationships.

Diameter = 2 × radius

or:

d = 2r

To find the radius from the diameter:

r = d ÷ 2

---

Example: Finding the Diameter

A circle has a radius of:

7 cm

Use:

d = 2r

Substitute:

d = 2 × 7

Therefore:

d = 14 cm

---

Example: Finding the Radius

A circle has a diameter of:

18 cm

Use:

r = d ÷ 2

Therefore:

r = 18 ÷ 2

r = 9 cm

---

The Circumference

The circumference is the distance around the outside of a circle.

For polygons, we usually call the distance around the shape the:

perimeter

For a circle, we use the special term:

circumference

Circumference is measured using units of length such as:

• Millimetres (mm).
• Centimetres (cm).
• Metres (m).
• Kilometres (km).

---

What Is π?

The symbol:

π

is the Greek letter pi.

Pi describes the relationship between the circumference of any circle and its diameter.

For every circle:

circumference ÷ diameter = π

Pi is approximately:

π ≈ 3.14

A calculator gives a more accurate value beginning:

π ≈ 3.14159265...

The decimal digits continue without ending or repeating in a fixed pattern.

For many basic calculations:

π ≈ 3.14

is accurate enough unless the question gives different instructions.

---

Why π Appears in Every Circle

Imagine measuring several circular objects.

For each object, measure:

• Its circumference.
• Its diameter.

Then calculate:

circumference ÷ diameter

You will obtain a value close to:

3.14

regardless of the size of the circle.

A small coin and a huge circular track have the same circumference-to-diameter ratio.

That ratio is:

π

---

Calculating Circumference Using Diameter

If the diameter is known:

C = πd

where:

C = circumference

d = diameter

and:

π ≈ 3.14

---

Worked Example: Circumference from Diameter

A circular plate has a diameter of:

20 cm

Use:

C = πd

Substitute:

C = π × 20

Using π ≈ 3.14:

C ≈ 3.14 × 20

C ≈ 62.8 cm

Therefore, the circumference is approximately:

62.8 cm

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Calculating Circumference Using Radius

Because:

d = 2r

we can replace the diameter in the circumference formula.

Therefore:

C = 2πr

where:

r = radius

This gives us two useful circumference formulas:

C = πd

or:

C = 2πr

The formula you choose depends on whether the problem gives you the radius or diameter.

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Exploring Radius, Diameter, and Circumference

Changing the radius changes both the diameter and circumference. The relationships remain the same for every circle.

Notice that doubling the radius also doubles:

• The diameter.
• The circumference.

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Worked Example: Circumference from Radius

A circle has a radius of:

6 cm

Use:

C = 2πr

Substitute:

C = 2 × π × 6

Using π ≈ 3.14:

C ≈ 2 × 3.14 × 6

C ≈ 37.68 cm

Therefore:

C ≈ 37.7 cm

to the nearest tenth.

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Choosing the Correct Formula

If the question gives the diameter, the simplest formula is:

C = πd

If the question gives the radius, the simplest formula is:

C = 2πr

You could always convert between radius and diameter first, but this is not necessary.

---

Example: Choosing the Formula

A bicycle wheel has a radius of:

35 cm

Since the radius is given, use:

C = 2πr

C = 2 × π × 35

C ≈ 219.9 cm

The wheel travels approximately:

220 cm

during one complete rotation, assuming it rolls without slipping.

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Estimating Circumference

Before calculating exactly, it is useful to estimate.

Since:

π ≈ 3

the circumference should be a little more than:

3 × diameter

Suppose:

d = 10 cm

Estimate:

C ≈ 3 × 10

C ≈ 30 cm

A more accurate calculation gives:

C ≈ 31.4 cm

The estimate tells us that the calculated answer is reasonable.

---

Example: Checking Reasonableness

Suppose a circle has:

d = 8 m

Someone calculates:

C = 2.51 m

This cannot be correct.

Since circumference is approximately three times the diameter, we expect:

C ≈ 3 × 8

≈ 24 m

The actual value is approximately:

25.1 m

Estimation helps detect calculation errors.

---

Finding Diameter from Circumference

Sometimes the circumference is known and the diameter must be calculated.

Starting with:

C = πd

Divide both sides by π:

d = C ÷ π

Example

A circular object has circumference:

31.4 cm

Calculate its diameter.

d = 31.4 ÷ 3.14

d = 10 cm

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Finding Radius from Circumference

Starting with:

C = 2πr

Rearrange:

r = C ÷ (2π)

Example

A circle has circumference:

62.8 cm

Using π ≈ 3.14:

r = 62.8 ÷ (2 × 3.14)

r = 62.8 ÷ 6.28

r = 10 cm

---

Exact and Approximate Answers

Sometimes circle calculations are left in terms of π.

For example:

A circle has:

r = 5 cm

Then:

C = 2π(5)

C = 10π cm

This is an exact answer.

Using π ≈ 3.14:

C ≈ 31.4 cm

This is an approximate answer.

The symbol:

≈

means:

approximately equal to

---

Why We Do Not Usually Write π = 3.14

Pi is not exactly 3.14.

Therefore:

π ≈ 3.14

is more accurate than writing:

π = 3.14

When using a calculator's π button, the calculator uses many more digits and produces a more accurate result.

---

Circumference and Units

Circumference measures length.

Therefore, circumference uses ordinary length units.

Examples:

25 cm

4.7 m

12 km

Do not use square units such as:

cm²

for circumference.

Square units are used for area, not circumference.

---

Radius, Diameter, and Circumference Compared

Important relationships:

d = 2r

r = d ÷ 2

C = πd

C = 2πr

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Measuring Real Circles

We can investigate circles using ordinary circular objects.

For example:

• Cup.
• Jar lid.
• Plate.
• Coin.
• Bottle.
• Wheel.

The diameter can be measured with a ruler.

The circumference can be measured by wrapping:

• String.
• Flexible measuring tape.

around the object.

---

Investigating π

Suppose we measure three circular objects.

Small differences can occur because of measurement error.

However, the ratio remains close to:

3.14

This provides experimental evidence for the relationship:

C = πd

---

Semicircles

A semicircle is half of a circle.

The curved part of a semicircle has a length equal to half the circumference.

If:

C = 2πr

then half the circumference is:

πr

However, the total perimeter of a semicircle also includes its straight diameter.

Therefore:

Perimeter of semicircle = πr + 2r

This distinction is important.

---

Example: Perimeter of a Semicircle

A semicircle has:

r = 4 cm

Curved part:

πr = 4π

Using π ≈ 3.14:

4π ≈ 12.56 cm

Diameter:

2r = 8 cm

Total perimeter:

12.56 + 8 = 20.56 cm

Therefore:

P ≈ 20.6 cm

---

Real-World Circles

Circular measurements are important in many everyday situations.

Examples include:

• Wheels.
• Tires.
• Gears.
• Clocks.
• Coins.
• Plates.
• Circular tables.
• Pipes.
• Storage tanks.
• Roundabouts.
• Running tracks.
• Ferris wheels.

---

Wheels and Circumference

Circumference is especially useful when studying wheels.

One complete rotation of a wheel moves the wheel forward by approximately one circumference, assuming it rolls without slipping.

Suppose a wheel has:

d = 0.6 m

Then:

C = πd

C ≈ 3.14 × 0.6

C ≈ 1.884 m

After 10 complete rotations:

distance ≈ 10 × 1.884

distance ≈ 18.84 m

---

Example: Bicycle Wheel

A bicycle wheel has a diameter of:

70 cm

Circumference:

C = πd

C ≈ 3.14 × 70

C ≈ 219.8 cm

Convert to metres:

219.8 cm = 2.198 m

Therefore, one complete rotation moves the bicycle approximately:

2.20 m

---

Circular Pipes

The diameter of pipes is an important engineering measurement.

A larger diameter affects:

• How much fluid can pass through.
• The size of fittings required.
• The circumference of the pipe.
• The amount of material needed around the pipe.

If a pipe has an outside diameter of:

12 cm

its outside circumference is:

C = π × 12

C ≈ 37.7 cm

---

Circular Tracks

Circumference can also be used to estimate distances around circular paths.

Suppose a circular track has:

r = 50 m

Then:

C = 2πr

C = 2 × π × 50

C = 100π

C ≈ 314 m

One complete lap is approximately:

314 m

---

Scaling Circles

Suppose the radius of a circle doubles.

Original radius:

r

New radius:

2r

Original circumference:

C = 2πr

New circumference:

C = 2π(2r)

C = 4πr

The new circumference is twice the original circumference.

Therefore:

Doubling the radius doubles the circumference.

Similarly:

Doubling the diameter doubles the circumference.

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A Reliable Circle Problem-Solving Strategy

When solving a circle problem:

Step 1: Identify what information is given.

Is it:

• Radius?
• Diameter?
• Circumference?

Step 2: Identify what you need to find.

Step 3: Choose the appropriate relationship.

d = 2r

r = d ÷ 2

C = πd

C = 2πr

Step 4: Substitute the known value.

Step 5: Calculate.

Step 6: Include the correct units.

Step 7: Estimate to check whether the answer is reasonable.

---

Worked Problem

A circular garden has a radius of:

8 m

Find its diameter and circumference.

Diameter

d = 2r

d = 2 × 8

d = 16 m

Circumference

C = 2πr

C = 2 × π × 8

C = 16π

Using π ≈ 3.14:

C ≈ 50.24 m

Therefore:

Diameter = 16 m

Circumference ≈ 50.2 m

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Worked Real-World Problem

A circular table has a diameter of:

1.5 m

A decorative strip will be placed around its outside edge.

How much strip is required?

Use:

C = πd

C = π × 1.5

C ≈ 4.71 m

Therefore, approximately:

4.71 m

of decorative strip is required.

In a real situation, slightly more material might be purchased to allow for joining and cutting.

---

Common Mistakes

Confusing Radius and Diameter

Remember:

radius = center to edge

diameter = edge to edge through the center

The diameter is twice the radius.

---

Forgetting to Double the Radius

If:

r = 7 cm

then:

d = 14 cm

not 7 cm.

---

Using C = πr

This is not the circumference formula.

Use:

C = πd

or:

C = 2πr

---

Multiplying the Diameter by 2 Again

If the diameter is already given, use:

C = πd

You do not need to double it.

---

Using the Wrong Units

Circumference is a length.

Use:

cm, m, km

not:

cm², m², km²

---

Treating π as Exactly 3.14

Remember:

π ≈ 3.14

When possible, use the calculator's π button for greater accuracy.

---

Rounding Too Early

Keep several digits during calculations and round the final answer.

Rounding too early can make the final answer less accurate.

---

Forgetting the Straight Side of a Semicircle

Half the circumference gives only the curved part.

The complete perimeter of a semicircle also includes its diameter.

---

Check Your Understanding

1. What is a circle?

2. What is the center of a circle?

3. Define radius.

4. Define diameter.

5. Define circumference.

6. What is the relationship between radius and diameter?

7. A circle has a radius of 6 cm. Find its diameter.

8. A circle has a radius of 11 m. Find its diameter.

9. A circle has a diameter of 16 cm. Find its radius.

10. A circle has a diameter of 25 m. Find its radius.

11. Approximately what value does π have?

12. Explain what π represents.

13. Write the circumference formula using diameter.

14. Write the circumference formula using radius.

15. Calculate the circumference of a circle with diameter 10 cm.

16. Calculate the circumference of a circle with diameter 24 cm.

17. Calculate the circumference of a circle with radius 5 cm.

18. Calculate the circumference of a circle with radius 12 m.

19. A circular table has diameter 2 m. Approximately how far is it around the edge?

20. A bicycle wheel has diameter 60 cm. Approximately how far does it travel during one complete rotation?

21. A circular track has radius 40 m. Find its circumference.

22. A circle has circumference 31.4 cm. Find its diameter using π ≈ 3.14.

23. A circle has circumference 62.8 cm. Find its radius.

24. Explain why circumference is measured in cm rather than cm².

25. Estimate the circumference of a circle with diameter 15 cm before calculating it accurately.

26. Explain how estimation can help identify mistakes in circumference calculations.

27. A wheel has a circumference of approximately 2 m. Approximately how far will it travel in 25 rotations?

28. A semicircle has radius 5 cm. Calculate the length of its curved edge.

29. Calculate the complete perimeter of the semicircle in Question 28.

30. Give three real-world situations in which radius, diameter, or circumference measurements are useful.

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Key Terms

• Circle – set of points in a plane that are all the same distance from a fixed center.
• Center – point exactly in the middle of a circle.
• Radius – distance from the center to the circumference.
• Diameter – distance across a circle through its center.
• Circumference – distance around a circle.
• Pi (π) – ratio of a circle's circumference to its diameter.
• Semicircle – half of a circle.
• Exact value – value written without approximation, such as 10π cm.
• Approximate value – rounded numerical value, such as 31.4 cm.

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Key Takeaways

• A circle is a two-dimensional curved shape.
• The center is the point in the middle.
• The radius runs from the center to the circumference.
• The diameter passes across the circle through the center.
• The diameter is twice the radius:

d = 2r

• The radius is half the diameter:

r = d ÷ 2

• The circumference is the distance around a circle.
• Pi is approximately:

π ≈ 3.14

• Circumference can be calculated using:

C = πd

or:

C = 2πr

• Circumference uses units of length, not square units.
• Circumference is slightly more than three times the diameter.
• Estimation is useful for checking whether an answer is reasonable.
• Radius, diameter, circumference, and π are used extensively in real-world measurement and design.

The four relationships worth remembering are:

d = 2r

r = d ÷ 2

C = πd

C = 2πr