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.