What Is Perimeter?

The perimeter of a two-dimensional shape is the total distance around the outside of the shape.

Imagine walking all the way around the edge of a playground. The total distance you walk is the perimeter of the playground.

To find perimeter:

Add the lengths of all the outside sides.

For any polygon:

Perimeter = sum of all side lengths

For example, if a triangle has side lengths:

4 cm, 6 cm, and 7 cm

then:

P = 4 + 6 + 7

P = 17 cm

Perimeter is a length, so it is measured in units such as:

• mm
• cm
• m
• km

Perimeter of a Rectangle

A rectangle has:

• Two equal lengths.
• Two equal widths.

 

 

 

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Suppose a rectangle has:

length = 8 cm

width = 5 cm

We could add all four sides:

P = 8 + 5 + 8 + 5

P = 26 cm

Because opposite sides are equal, we can use:

P = 2l + 2w

or:

P = 2(l + w)

where:

P = perimeter

l = length

w = width

Worked Example

A rectangular garden is:

12 m long

and:

7 m wide

Find its perimeter.

P = 2(l + w)

P = 2(12 + 7)

P = 2(19)

P = 38 m

The garden has a perimeter of 38 m.

Perimeter of a Square

A square has four equal sides.

 

 

 

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If each side has length s:

P = 4s

Worked Example

A square has a side length of:

6 cm

Therefore:

P = 4 × 6

P = 24 cm

Perimeter of a Triangle

A triangle has three sides.

To find its perimeter:

P = a + b + c

Worked Example

A triangle has side lengths:

5 cm, 8 cm, and 10 cm

Therefore:

P = 5 + 8 + 10

P = 23 cm

Notice that the sides do not need to be equal.

Perimeter of Other Polygons

The same basic rule works for any polygon:

Add every outside side.

Suppose a pentagon has side lengths:

3 cm, 5 cm, 4 cm, 6 cm, and 7 cm

Then:

P = 3 + 5 + 4 + 6 + 7

P = 25 cm

This method works for:

• Triangles.
• Quadrilaterals.
• Pentagons.
• Hexagons.
• Regular polygons.
• Irregular polygons.

Regular Polygons

A regular polygon has sides of equal length.

If a regular polygon has:

n sides

and each side has length:

s

then:

P = ns

Example: Regular Pentagon

A regular pentagon has five sides.

Each side is:

7 cm

Therefore:

P = 5 × 7

P = 35 cm

Example: Regular Octagon

A regular octagon has eight equal sides.

If each side is:

4.5 m

then:

P = 8 × 4.5

P = 36 m

Irregular Polygons

An irregular polygon does not have all sides equal.

There may not be a shortcut formula.

Instead:

Add all the outside side lengths.

Example

An irregular shape has side lengths:

4 m, 7 m, 3 m, 5 m, 6 m, and 8 m

Therefore:

P = 4 + 7 + 3 + 5 + 6 + 8

P = 33 m

Finding Missing Side Lengths

Sometimes a diagram does not give every side length directly.

You may need to determine a missing side before calculating the perimeter.

Example

A rectangle has:

length = 11 cm

width = 4 cm

Only three sides are labelled:

11 cm, 4 cm, 11 cm

What is the missing side?

Opposite sides of a rectangle are equal.

Therefore:

missing side = 4 cm

Now:

P = 11 + 4 + 11 + 4

P = 30 cm

Finding a Missing Side from the Perimeter

Sometimes the perimeter is known, but one side is missing.

Worked Example

A triangle has a perimeter of:

30 cm

Two sides measure:

8 cm and 12 cm

Find the third side.

Let the missing side be x.

8 + 12 + x = 30

20 + x = 30

x = 10 cm

The missing side is 10 cm.

Another Missing-Side Example

A rectangle has a perimeter of:

42 m

Its length is:

13 m

Find its width.

Use:

P = 2l + 2w

Substitute the known values:

42 = 2(13) + 2w

42 = 26 + 2w

Subtract 26:

16 = 2w

Divide by 2:

w = 8 m

The rectangle is 8 m wide.

Compound Shapes

A compound shape is made by combining simpler shapes.

 

 

 

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When calculating its perimeter, include only the sides around the outside boundary.

Do not include internal lines.

A useful strategy is:

1. Start at one corner.
2. Move around the outside of the shape.
3. Write down each side length.
4. Determine any missing lengths.
5. Add the outside lengths.
6. Check that you have returned to your starting point.

Finding Missing Lengths in Compound Shapes

Consider an L-shaped figure.

Suppose its total height is:

10 cm

One vertical section is:

6 cm

The missing vertical section must be:

10 − 6 = 4 cm

This missing length can then be included when calculating the perimeter.

A similar method works horizontally.

If the total width is:

15 cm

and one horizontal section is:

9 cm

then the missing section is:

15 − 9 = 6 cm

Perimeter in Real Life

Perimeter is useful whenever we need to know the distance around something.

Examples include:

• Fencing a garden.
• Building a fence around a property.
• Installing a border around a garden bed.
• Putting trim around a window.
• Framing a picture.
• Marking the boundary of a sports field.
• Installing baseboards around a room.
• Decorating the edge of a poster.

The important question is:

Do I need to know the distance around the object?

If yes, perimeter is probably involved.

Worked Example: Fencing a Garden

A rectangular garden is:

15 m long

and:

9 m wide

How much fencing is needed to completely surround it?

P = 2(l + w)

P = 2(15 + 9)

P = 48 m

Therefore, 48 m of fencing is required.

Worked Example: Cost of Fencing

Suppose fencing costs:

$12 per metre

The garden has a perimeter of:

48 m

Cost:

48 × $12 = $576

Therefore, the fencing costs:

$576

This is a common real-world use of perimeter.

Worked Example: Picture Frame

A photograph is:

30 cm by 20 cm

How much framing material is needed?

P = 2(30 + 20)

P = 2(50)

P = 100 cm

Therefore, 100 cm of framing material is required.

This could also be written as:

1 m

Perimeter and Units

Because perimeter measures distance, we use linear units.

Correct examples:

P = 25 cm

P = 18 m

P = 3.4 km

We do not use square units for perimeter.

Incorrect:

P = 25 cm²

Correct:

P = 25 cm

Square units such as cm² and m² are used for area.

Converting Units

Sometimes side lengths are given in different units.

Convert them to the same unit before adding.

Example

A rectangle has:

length = 2 m

width = 50 cm

We should not calculate:

2 + 50

because the units are different.

Convert:

2 m = 200 cm

Then:

P = 2(200 + 50)

P = 500 cm

or:

P = 5 m

Perimeter vs Area

Perimeter and area describe different properties of a shape.

Perimeter

Measures the distance around a shape.

Area

Measures the amount of surface inside a shape.

 

 

 

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Think of a garden.

Fence around the garden → perimeter

Grass covering the garden → area

Comparing Perimeter and Area

Worked Example: Perimeter and Area

A rectangle measures:

8 m × 5 m

Perimeter

P = 2(8 + 5)

P = 26 m

Area

A = 8 × 5

A = 40 m²

Notice the different units:

Perimeter = 26 m

Area = 40 m²

They represent completely different measurements.

Shapes Can Have the Same Perimeter but Different Areas

Consider two rectangles.

Rectangle A

8 cm × 2 cm

Perimeter:

P = 2(8 + 2) = 20 cm

Area:

A = 8 × 2 = 16 cm²

Rectangle B

5 cm × 5 cm

Perimeter:

P = 4 × 5 = 20 cm

Area:

A = 5 × 5 = 25 cm²

Both shapes have the same perimeter:

20 cm

but their areas are different.

This demonstrates that knowing the perimeter does not necessarily tell you the area.

Shapes Can Have the Same Area but Different Perimeters

Consider:

Rectangle A

12 cm × 2 cm

Area:

24 cm²

Perimeter:

28 cm

Rectangle B

6 cm × 4 cm

Area:

24 cm²

Perimeter:

20 cm

Both have the same area but different perimeters.

Therefore:

same area ≠ same perimeter

and:

same perimeter ≠ same area

Finding a Missing Dimension from Perimeter

Perimeter can be used to work backward.

Example

A rectangular field has a perimeter of:

100 m

Its length is:

30 m

Find its width.

P = 2l + 2w

100 = 2(30) + 2w

100 = 60 + 2w

40 = 2w

w = 20 m

Real-World Example: Running Track

A student runs around the outside of a rectangular field measuring:

80 m × 50 m

One lap is:

P = 2(80 + 50)

P = 260 m

If the student completes four laps:

distance = 4 × 260

distance = 1040 m

Therefore, the student runs:

1.04 km

Real-World Example: Garden Border

A square garden has sides of:

6.5 m

The owner wants to install a decorative border around the entire garden.

P = 4 × 6.5

P = 26 m

If the border material comes in 2 m sections:

26 ÷ 2 = 13

Therefore, 13 sections are required.

Real-World Example: A Gate in a Fence

Sometimes part of the perimeter does not require the same material.

A rectangular yard measures:

20 m × 12 m

Total perimeter:

P = 2(20 + 12)

P = 64 m

A gate takes up:

3 m

Therefore, fencing required:

64 − 3 = 61 m

This shows why understanding the situation is important instead of automatically using a formula and stopping.

A Reliable Problem-Solving Strategy

When solving a perimeter problem:

Step 1: Identify the shape.

Step 2: Identify all known side lengths.

Step 3: Determine any missing lengths.

Step 4: Convert units if necessary.

Step 5: Add all outside sides or use an appropriate formula.

Step 6: Include the correct linear unit.

Step 7: Ask whether the answer is reasonable.

Estimating Perimeter

Estimation can help check your calculations.

Suppose a rectangle measures:

19.8 m × 10.2 m

Estimate:

20 m × 10 m

Estimated perimeter:

2(20 + 10) = 60 m

Exact calculation:

P = 2(19.8 + 10.2)

P = 2(30)

P = 60 m

The estimate confirms that the answer is reasonable.

Common Mistakes

Confusing Perimeter and Area

Remember:

Perimeter = around

Area = inside

Using Square Units

Incorrect:

P = 32 cm²

Correct:

P = 32 cm

Perimeter is a length.

Forgetting a Side

For an irregular or compound shape, trace around the entire outside boundary.

Make sure every outside side is included.

Counting Internal Lines

Only the outside boundary contributes to perimeter.

Lines inside a compound shape should not be included.

Assuming All Sides Are Equal

A square has four equal sides.

A rectangle has equal opposite sides.

An irregular polygon may have no equal sides.

Adding Different Units

Do not calculate:

2 m + 40 cm = 42

Convert to the same unit first.

For example:

2 m = 200 cm

so:

200 cm + 40 cm = 240 cm

Multiplying Instead of Adding

For a rectangle:

8 × 5 = 40

calculates the area, not the perimeter.

Perimeter:

2(8 + 5) = 26

Forgetting to Determine Missing Sides

Some compound-shape problems intentionally leave side lengths unlabelled.

Use the geometry of the shape to determine them before calculating the perimeter.

Check Your Understanding

1. Define perimeter.

2. What units are used for perimeter?

3. Find the perimeter of a square with side length 9 cm.

4. Find the perimeter of a rectangle measuring 12 cm by 7 cm.

5. Find the perimeter of a triangle with sides 6 cm, 8 cm, and 11 cm.

6. A regular pentagon has sides of 5 cm. Find its perimeter.

7. A regular hexagon has sides of 7.5 cm. Find its perimeter.

8. An irregular polygon has side lengths 3 m, 7 m, 4 m, 8 m, and 6 m. Find its perimeter.

9. A triangle has a perimeter of 40 cm. Two sides measure 12 cm and 15 cm. Find the third side.

10. A rectangle has a perimeter of 50 cm and a length of 15 cm. Find its width.

11. A square has a perimeter of 36 m. Find its side length.

12. A rectangular garden measures 18 m by 11 m. How much fencing is required?

13. The fencing in Question 12 costs $8 per metre. Find the total cost.

14. A picture measures 40 cm by 25 cm. How much framing material is required?

15. Explain the difference between perimeter and area.

16. What units would you use for the area of a classroom? What units would you use for its perimeter?

17. A rectangle measures 10 cm × 4 cm. Calculate both its perimeter and area.

18. A rectangular field measures 100 m × 60 m. A runner completes three laps around its edge. How far does the runner travel?

19. A rectangular yard has a perimeter of 70 m and a width of 12 m. Find its length.

20. A rectangle measures 2 m by 75 cm. Find its perimeter in metres.

21. Explain why 35 cm² cannot be a correct unit for perimeter.

22. Two rectangles have the same perimeter. Must they have the same area? Explain using an example.

23. Two rectangles have the same area. Must they have the same perimeter? Explain.

24. A square garden has a perimeter of 48 m. Find its area.

25. A rectangular garden measures 14 m × 8 m. A 2 m-wide gate is installed. How many metres of fencing are required for the rest of the boundary?

Key Terms

• Perimeter – total distance around a two-dimensional shape.
• Polygon – closed two-dimensional shape made from straight sides.
• Regular polygon – polygon with equal side lengths and equal angles.
• Irregular polygon – polygon whose sides and/or angles are not all equal.
• Rectangle – quadrilateral with four right angles and equal opposite sides.
• Square – quadrilateral with four equal sides and four right angles.
• Compound shape – shape made from two or more simpler shapes.
• Side length – distance from one vertex of a polygon to the next.
• Area – amount of two-dimensional space inside a shape.
• Linear unit – unit used to measure length, such as cm or m.
• Square unit – unit used to measure area, such as cm² or m².

Key Takeaways

• Perimeter is the total distance around a two-dimensional shape.
• To find the perimeter of any polygon, add all the outside side lengths.
• Rectangle: P = 2(l + w)
• Square: P = 4s
• Regular polygon: P = number of sides × side length
• Missing side lengths may need to be determined before calculating perimeter.
• Opposite sides of a rectangle are equal.
• Compound shapes require careful tracing of the entire outside boundary.
• Internal lines are not included in perimeter.
• All measurements must use compatible units before they are added.
• Perimeter is measured in linear units such as cm, m, and km.
• Area is measured in square units such as cm² and m².
• Perimeter measures the distance around a shape.
• Area measures the space inside a shape.
• Shapes can have the same perimeter but different areas.
• Shapes can have the same area but different perimeters.
• Perimeter has many practical applications involving fencing, borders, frames, trim, and travel around boundaries.

A simple way to remember the difference is:

Perimeter = around.

Area = inside.

The formula is:

A = ½ × b × h

where:

A = area

b = base

h = perpendicular height

For example, if:

b = 8 cm

h = 6 cm

then:

A = ½ × 8 × 6

A = 24 cm²

Identifying the Base

The base is the side of the triangle that you choose to use when calculating its area.

A triangle does not have only one possible base.

Any of its three sides can be chosen as the base, provided you use the correct perpendicular height for that base.

 

 

 

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For most simple problems, the base is drawn horizontally along the bottom of the triangle.

However, this does not always happen.

Identifying the Height

The height is the perpendicular distance from the base to the opposite vertex.

Perpendicular means that the height meets the base at an angle of:

90°

This is extremely important.

A sloping side of the triangle is not necessarily the height.

Right Triangles

In a right triangle, the two sides that meet at the right angle can be used as the base and height.

Suppose a right triangle has perpendicular sides:

10 cm and 7 cm

Then:

A = ½ × 10 × 7

A = 35 cm²

The third side is not needed.

Worked Example: Basic Triangle

A triangle has:

base = 12 cm

height = 7 cm

Calculate its area.

Use:

A = ½ × b × h

Substitute:

A = ½ × 12 × 7

A = 42 cm²

Therefore, the area is:

42 cm²

Why Do We Divide by Two?

Imagine a rectangle measuring:

8 cm × 5 cm

Its area is:

8 × 5 = 40 cm²

If the rectangle is divided diagonally into two identical triangles, each triangle occupies half of the rectangle.

Therefore:

40 ÷ 2 = 20 cm²

This explains the ½ in the triangle-area formula.

 

 

 

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The Height Can Be Inside the Triangle

For many triangles, the perpendicular height lies inside the triangle.

For example:

base = 14 m

height = 9 m

Then:

A = ½ × 14 × 9

A = 63 m²

The Height Can Be Outside the Triangle

For an obtuse triangle, the perpendicular height may fall outside the triangle.

The base can be extended with an imaginary line until the perpendicular height meets it.

The same formula still works:

A = ½ × b × h

The important idea is that the height must be measured perpendicular to the chosen base.

Finding a Missing Height

Sometimes the area and base are known.

Suppose:

A = 45 cm²

b = 10 cm

Use:

A = ½ × b × h

Substitute:

45 = ½ × 10 × h

45 = 5h

Therefore:

h = 9 cm

Finding a Missing Base

Suppose:

A = 60 m²

h = 8 m

Use:

A = ½ × b × h

Substitute:

60 = ½ × b × 8

60 = 4b

Therefore:

b = 15 m

A Useful Rearrangement

Because:

A = ½bh

we can also write:

2A = bh

Therefore:

b = 2A ÷ h

and:

h = 2A ÷ b

These forms are useful when finding missing dimensions.

Units of Area

Area is measured in square units.

Examples include:

• mm²
• cm²
• m²
• km²

If the base and height are measured in centimetres, the area will usually be measured in:

cm²

If they are measured in metres:

m²

Composite Shapes

A composite shape is a shape made by combining two or more simpler shapes.

For example, a composite figure might contain:

• Rectangles.
• Squares.
• Triangles.

 

 

 

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Instead of looking for one complicated formula, divide the figure into shapes whose areas you already know how to calculate.

The Basic Composite-Area Strategy

To calculate the area of a composite shape:

Step 1: Identify the simpler shapes.

Step 2: Divide the composite figure into those shapes.

Step 3: Determine any missing dimensions.

Step 4: Calculate each individual area.

Step 5: Add the areas together.

In some problems, it is easier to calculate the area of a large shape and subtract a missing section.

Adding Areas

Suppose a composite figure contains:

Rectangle

8 cm × 5 cm

Area:

A = 8 × 5

A = 40 cm²

Triangle

base = 8 cm

height = 4 cm

Area:

A = ½ × 8 × 4

A = 16 cm²

Total area:

40 + 16 = 56 cm²

Worked Example: House-Shaped Figure

Imagine a simple house shape consisting of:

• A rectangular wall section.
• A triangular roof.

The rectangle measures:

10 m × 6 m

The triangular roof has:

base = 10 m

height = 3 m

Rectangle

A = 10 × 6

A = 60 m²

Triangle

A = ½ × 10 × 3

A = 15 m²

Total Area

60 + 15 = 75 m²

Therefore:

Total area = 75 m²

Finding Missing Dimensions in Composite Shapes

Composite-shape diagrams may not provide every measurement directly.

Suppose the total width is:

15 cm

One section has a width of:

9 cm

The missing width is:

15 − 9 = 6 cm

Similarly, if the total height is:

12 cm

and one section has height:

7 cm

the missing height is:

12 − 7 = 5 cm

Find these missing dimensions before calculating the individual areas.

L-Shaped Figures

An L-shaped figure can often be divided into two rectangles.

 

 

 

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Suppose:

Rectangle A

8 cm × 4 cm

Area:

32 cm²

Rectangle B

3 cm × 5 cm

Area:

15 cm²

Total:

32 + 15 = 47 cm²

Therefore:

Total area = 47 cm²

More Than One Method

Composite shapes can often be divided in different ways.

An L-shaped figure might be divided:

vertically

or:

horizontally

Both methods should produce the same total area.

This gives you a useful way to check your work.

Subtracting a Missing Section

Sometimes subtraction is easier than dividing the shape into several pieces.

Imagine a large rectangle measuring:

12 cm × 10 cm

with a smaller rectangle measuring:

4 cm × 3 cm

removed from one corner.

Large Rectangle

A = 12 × 10

A = 120 cm²

Missing Rectangle

A = 4 × 3

A = 12 cm²

Remaining Area

120 − 12 = 108 cm²

Therefore:

Area = 108 cm²

Add or Subtract?

For composite shapes, ask:

Would it be easier to divide the figure into pieces and add them?

or:

Would it be easier to imagine a larger complete shape and subtract the missing piece?

Either method is valid.

Choose the method that makes the calculation simplest.

Composite Shape with a Triangle

Suppose a shape consists of:

• A rectangle measuring 12 cm × 6 cm
• A triangle on top with base = 12 cm and height = 5 cm

Rectangle:

A = 12 × 6 = 72 cm²

Triangle:

A = ½ × 12 × 5 = 30 cm²

Total:

A = 72 + 30

A = 102 cm²

Composite Shape with Two Triangles

Suppose a rectangle measures:

10 cm × 6 cm

A triangle with:

base = 4 cm

height = 3 cm

is attached to each end.

Rectangle:

A = 10 × 6

A = 60 cm²

Each triangle:

A = ½ × 4 × 3

A = 6 cm²

Two triangles:

2 × 6 = 12 cm²

Total:

60 + 12 = 72 cm²

Irregular Shapes

Some real-world objects do not form perfect rectangles, squares, or triangles.

One approach is to approximate the irregular shape using simpler geometric shapes.

For example, an irregular garden might be divided into:

• Two rectangles.
• One triangle.

Calculate each area and add them.

The answer may be an approximation if the actual boundary is curved or irregular.

Real-World Example: Triangular Garden

A triangular garden has:

base = 16 m

perpendicular height = 9 m

Area:

A = ½ × 16 × 9

A = 72 m²

If grass seed covers:

12 m² per bag

then:

72 ÷ 12 = 6

Therefore:

6 bags of grass seed

are required.

Real-World Example: Painting a Wall

A wall consists of a rectangle topped by a triangle.

Rectangle:

8 m × 3 m

Triangle:

base = 8 m

height = 2 m

Rectangle area:

8 × 3 = 24 m²

Triangle area:

½ × 8 × 2 = 8 m²

Total wall area:

24 + 8 = 32 m²

If one litre of paint covers:

8 m²

then:

32 ÷ 8 = 4

Approximately 4 litres of paint are required for one coat.

Real-World Example: Flooring

An L-shaped room can be divided into two rectangles.

Suppose:

Rectangle A = 6 m × 4 m

Rectangle B = 3 m × 2 m

Calculate:

Area A = 24 m²

Area B = 6 m²

Therefore:

Total floor area = 30 m²

If flooring costs:

$25 per m²

then:

30 × 25 = $750

The basic flooring cost is:

$750

Real-World Example: Land

A piece of land can be approximated as:

• A rectangle measuring 30 m × 20 m
• A triangle measuring 20 m base × 12 m height

Rectangle:

A = 30 × 20 = 600 m²

Triangle:

A = ½ × 20 × 12 = 120 m²

Total:

A = 600 + 120

A = 720 m²

Real-World Example: Subtracting a Pool

A rectangular backyard measures:

20 m × 15 m

A rectangular swimming pool measures:

8 m × 5 m

How much land remains outside the pool?

Backyard:

A = 20 × 15

A = 300 m²

Pool:

A = 8 × 5

A = 40 m²

Remaining area:

300 − 40 = 260 m²

Real-World Example: Pathway

A rectangular garden measures:

12 m × 8 m

Inside it is a rectangular planted area measuring:

10 m × 6 m

The remaining area forms a pathway.

Total garden:

A = 12 × 8 = 96 m²

Planted area:

A = 10 × 6 = 60 m²

Pathway:

A = 96 − 60

A = 36 m²

Perimeter Is Not Area

Composite-shape problems can involve either area or perimeter.

Do not confuse them.

Area

Measures the space inside.

Measured in:

cm², m², km²

Perimeter

Measures the distance around the outside.

Measured in:

cm, m, km

If a question asks how much:

• Carpet.
• Paint.
• Grass.
• Tile.
• Material covering.

you probably need area.

If it asks about:

• Fencing.
• Border.
• Edging.
• Distance around.

you probably need perimeter.

Estimating Before Calculating

Estimation can help identify mistakes.

Suppose a triangle has:

base = 19.8 m

height = 10.2 m

Estimate:

½ × 20 × 10

≈ 100 m²

Exact calculation:

A = ½ × 19.8 × 10.2

A = 100.98 m²

The exact answer is close to the estimate, so it appears reasonable.

A Reliable Problem-Solving Method

For a triangle:

1. Identify the base.
2. Identify the perpendicular height.
3. Check that the units match.
4. Use A = ½ × b × h.
5. Calculate.
6. Include square units.

For a composite shape:

1. Examine the entire figure.
2. Divide it into familiar shapes.
3. Determine missing dimensions.
4. Calculate each area separately.
5. Add or subtract the areas as required.
6. Include square units.
7. Check whether the result is reasonable.

Common Mistakes

Forgetting to Divide by Two

Incorrect:

A = b × h

for a triangle.

Correct:

A = ½ × b × h

Using a Sloping Side as the Height

The height must be perpendicular to the base.

Look for a 90° relationship.

Using the Wrong Height

A triangle can have different bases, but each base has its own corresponding perpendicular height.

Using Perimeter Instead of Area

Adding all the side lengths gives the perimeter, not the area.

Forgetting Square Units

Incorrect:

A = 35 cm

Correct:

A = 35 cm²

Adding Lengths Before Calculating Area

For composite figures, calculate the area of each section first.

Then add the areas.

Do not simply add all the side lengths.

Including a Missing Section

If part of a larger rectangle has been removed, its area must be subtracted.

Missing Part of a Composite Shape

Check that every part of the figure has been included exactly once.

Counting the Same Region Twice

When dividing a composite shape, make sure the simpler shapes do not overlap.

Otherwise, part of the area will be counted twice.

Ignoring Different Units

If one measurement is in metres and another is in centimetres, convert them to compatible units before calculating.

Check Your Understanding

1. Write the formula for the area of a triangle.

2. What is meant by the base of a triangle?

3. What is meant by the perpendicular height?

4. Why can't any sloping side automatically be used as the height?

5. Find the area of a triangle with base 10 cm and height 6 cm.

6. Find the area of a triangle with base 15 m and height 8 m.

7. A triangle has a base of 12 cm and a height of 9 cm. Find its area.

8. A right triangle has perpendicular sides of 7 cm and 11 cm. Find its area.

9. A triangle has an area of 40 cm² and a base of 10 cm. Find its height.

10. A triangle has an area of 54 m² and a height of 9 m. Find its base.

11. Explain why the area of a triangle is half the area of a rectangle with the same base and height.

12. A composite figure consists of a 10 cm × 6 cm rectangle and a triangle with base 10 cm and height 4 cm. Find the total area.

13. A house-shaped figure contains a rectangle measuring 12 m × 5 m and a triangular roof with base 12 m and height 3 m. Find its total area.

14. An L-shaped figure is divided into rectangles with areas of 30 cm² and 18 cm². Find the total area.

15. A 15 cm × 10 cm rectangle has a 5 cm × 4 cm rectangle removed. Find the remaining area.

16. A triangular garden has a base of 20 m and a perpendicular height of 12 m. Find its area.

17. Grass seed covers 15 m² per bag. How many bags are needed for the garden in Question 16?

18. A wall consists of a rectangle measuring 9 m × 3 m and a triangular section with base 9 m and height 2 m. Find its total area.

19. One litre of paint covers 6 m². How many litres are required for one coat on the wall in Question 18?

20. Explain two different strategies that can be used to calculate the area of an L-shaped figure.

21. Why should you determine missing dimensions before calculating the areas of some composite shapes?

22. A rectangular yard measures 18 m × 12 m. A rectangular pool measuring 7 m × 4 m occupies part of it. Find the area of the yard not occupied by the pool.

23. A triangle has an area of 75 cm² and a height of 10 cm. Find its base.

24. Explain the difference between area and perimeter.

25. Describe the steps you would use to calculate the area of a complicated figure made from several rectangles and triangles.

Key Terms

• Area – amount of two-dimensional space inside a shape.
• Triangle – polygon with three sides.
• Base – chosen side of a triangle used when calculating its area.
• Height – perpendicular distance from a base to the opposite vertex.
• Perpendicular – meeting at a 90° angle.
• Composite shape – figure made from two or more simpler shapes.
• Compound shape – another term commonly used for a composite shape.
• Rectangle – four-sided shape with four right angles and equal opposite sides.
• Square – four-sided shape with four equal sides and four right angles.
• Square unit – unit used for measuring area, such as cm² or m².
• Perimeter – total distance around a shape.

Key Takeaways

• The area of a triangle depends on its base and perpendicular height.
• The triangle-area formula is A = ½ × b × h.
• The height must meet the chosen base at 90°.
• A sloping side is not automatically the height.
• Any side of a triangle can serve as the base if the correct corresponding height is used.
• In a right triangle, the two perpendicular sides can serve as the base and height.
• Area is measured in square units such as cm² and m².
• Composite shapes can be divided into simpler shapes.
• Calculate the area of each simpler shape separately.
• Add the areas when pieces form the complete figure.
• Subtract an area when a section has been removed.
• Missing dimensions may need to be calculated before finding the area.
• Different methods of dividing a composite shape should produce the same total area.
• Estimation is useful for checking whether an answer is reasonable.
• Area calculations have practical applications in flooring, painting, landscaping, construction, and land measurement.

A useful strategy for composite shapes is:

Break it apart → calculate each area → add or subtract → check the units.

What Is Surface Area?

Surface area is the total area of all the outside surfaces, or faces, of a three-dimensional object.

Imagine wrapping a box completely with paper. The amount of paper needed to cover every outside face of the box represents its surface area.

Surface area is useful when determining:

• How much cardboard is needed to make a box.
• How much wrapping paper is needed for a package.
• How much paint is needed to cover an object.
• How much material is needed to manufacture a container.
• How much exposed surface an object has.

 

 

 

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Because surface area measures area, it is measured in square units such as:

cm², m², and km²

Faces of Three-Dimensional Objects

A face is a flat surface of a three-dimensional object.

A rectangular prism has:

• 6 faces
• 12 edges
• 8 vertices

Its opposite faces are identical.

A rectangular prism has three pairs of faces:

• Top and bottom.
• Front and back.
• Left and right.

This is important when calculating surface area because each type of face occurs twice.

Rectangular Prisms

A rectangular prism has three dimensions:

length (l)

width (w)

height

Its six faces consist of:

2 faces measuring l × w

2 faces measuring l × h

2 faces measuring w × h

Therefore:

SA = 2lw + 2lh + 2wh

This can also be written:

SA = 2(lw + lh + wh)

where:

SA = surface area

Understanding the Formula

Suppose a rectangular prism measures:

8 cm × 5 cm × 3 cm

There are three different face areas.

Top and Bottom

Each measures:

8 × 5 = 40 cm²

There are two:

2 × 40 = 80 cm²

Front and Back

Each measures:

8 × 3 = 24 cm²

There are two:

2 × 24 = 48 cm²

Left and Right

Each measures:

5 × 3 = 15 cm²

There are two:

2 × 15 = 30 cm²

Now add:

SA = 80 + 48 + 30

SA = 158 cm²

A Face-by-Face Method

You do not have to memorize the formula immediately.

You can calculate each face separately.

For a rectangular prism:

top + bottom + front + back + left + right

Then add all six areas.

This method can be especially helpful when first learning surface area.

Nets

A net is a two-dimensional pattern that can be folded to make a three-dimensional object.

 

 

 

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Imagine cutting along some edges of a cardboard box and unfolding it until it lies flat.

You would see all six faces.

The net makes it easier to see that surface area is simply:

the sum of the areas of all the faces

Nets of Rectangular Prisms

A rectangular prism can have several different nets.

Although the arrangement of the rectangles may change, every valid net contains the same six faces.

For a prism with dimensions:

l × w × h

the net contains:

• Two rectangles measuring l × w
• Two rectangles measuring l × h
• Two rectangles measuring w × h

Therefore:

SA = 2lw + 2lh + 2wh

Worked Example: Rectangular Prism

A box measures:

10 cm long

6 cm wide

4 cm high

Find its surface area.

Use:

SA = 2lw + 2lh + 2wh

Substitute:

SA = 2(10 × 6) + 2(10 × 4) + 2(6 × 4)

Calculate:

SA = 2(60) + 2(40) + 2(24)

SA = 120 + 80 + 48

SA = 248 cm²

Therefore:

Surface area = 248 cm²

Another Worked Example

A rectangular prism measures:

12 m × 5 m × 3 m

Calculate the three face areas:

12 × 5 = 60 m²

12 × 3 = 36 m²

5 × 3 = 15 m²

Each occurs twice:

SA = 2(60) + 2(36) + 2(15)

SA = 120 + 72 + 30

SA = 222 m²

Cubes

A cube is a special rectangular prism.

All six faces of a cube are identical squares.

 

 

 

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If the side length is:

s

the area of one face is:

s²

Since a cube has six faces:

SA = 6s²

Worked Example: Cube

A cube has side length:

5 cm

Area of one face:

5 × 5 = 25 cm²

There are six faces:

SA = 6 × 25

SA = 150 cm²

Therefore:

Surface area = 150 cm²

Another Cube Example

A cube has sides measuring:

8 m

Use:

SA = 6s²

SA = 6 × 8²

SA = 6 × 64

SA = 384 m²

Surface Area vs Area

Area describes the size of a two-dimensional surface.

For example, a rectangle measuring:

8 cm × 5 cm

has area:

40 cm²

Surface area combines the areas of all the outside surfaces of a three-dimensional object.

So surface area is essentially:

the total area of all exposed faces

Surface Area vs Volume

Surface area and volume describe very different properties.

Surface Area

Measures the amount of material covering the outside.

Measured in:

cm² or m²

Volume

Measures the amount of space inside.

Measured in:

cm³ or m³

Consider a cardboard box.

Cardboard required → surface area

Space inside the box → volume

Comparing Area, Surface Area, and Volume

Using Nets to Calculate Surface Area

Suppose a net contains six rectangles with these areas:

24 cm²

24 cm²

18 cm²

18 cm²

12 cm²

12 cm²

Add them:

SA = 24 + 24 + 18 + 18 + 12 + 12

SA = 108 cm²

The folded prism therefore has a surface area of:

108 cm²

Missing Faces

Not every real-world surface-area problem includes all six faces.

For example, an open-top box has only five exposed faces.

Suppose a rectangular box measures:

10 cm × 6 cm × 4 cm

A closed box has:

SA = 248 cm²

The top has area:

10 × 6 = 60 cm²

If the top is missing:

248 − 60 = 188 cm²

Therefore:

Surface area of the open box = 188 cm²

This is why understanding the object is more important than simply applying a formula.

Open Containers

Consider an open rectangular storage container.

To calculate the amount of material required, include:

• The bottom.
• The front.
• The back.
• The left side.
• The right side.

Do not include the top if no top exists.

A diagram or net can help identify exactly which surfaces should be counted.

Practical Example: Wrapping a Box

A gift box measures:

30 cm × 20 cm × 10 cm

Calculate its surface area.

SA = 2(lw + lh + wh)

SA = 2[(30 × 20) + (30 × 10) + (20 × 10)]

SA = 2(600 + 300 + 200)

SA = 2(1100)

SA = 2200 cm²

At least:

2200 cm²

of paper is required to cover the surfaces exactly.

In reality, more wrapping paper would be needed for overlap and folding.

Practical Example: Painting a Box

A large rectangular structure measures:

4 m × 3 m × 2 m

Find the total outside surface area.

SA = 2[(4 × 3) + (4 × 2) + (3 × 2)]

SA = 2(12 + 8 + 6)

SA = 52 m²

If one litre of paint covers:

8 m²

then:

52 ÷ 8 = 6.5

The theoretical paint requirement is:

6.5 L

In practice, the amount needed may depend on the surface, number of coats, and waste.

Practical Example: Packaging

A company wants to make a rectangular cardboard box measuring:

40 cm × 30 cm × 20 cm

Calculate the minimum cardboard area required for a completely closed box.

SA = 2[(40 × 30) + (40 × 20) + (30 × 20)]

SA = 2(1200 + 800 + 600)

SA = 5200 cm²

Therefore, the six faces require:

5200 cm² of cardboard

before allowing for tabs, folds, seams, and waste.

Practical Example: An Open Aquarium

An aquarium has dimensions:

80 cm × 30 cm × 40 cm

Suppose it has:

• A bottom.
• Four sides.
• No top.

Bottom:

80 × 30 = 2400 cm²

Front and back:

2(80 × 40) = 6400 cm²

Two ends:

2(30 × 40) = 2400 cm²

Total:

SA = 2400 + 6400 + 2400

SA = 11,200 cm²

Therefore, the glass area required is:

11,200 cm²

before considering overlaps, thickness, or construction details.

Practical Example: Painting a Room

A room is also a rectangular prism, but we usually do not paint all six surfaces.

Suppose a room measures:

5 m long

4 m wide

3 m high

If we want to paint only the four walls:

Two long walls:

2(5 × 3) = 30 m²

Two short walls:

2(4 × 3) = 24 m²

Total wall area:

30 + 24 = 54 m²

We would not include the floor or ceiling unless the problem specifically required them.

Subtracting Doors and Windows

Real-world surface-area problems may include sections that should not be covered.

Suppose the walls of a room have an area of:

54 m²

A door has an area of:

2 m²

and the windows have a total area of:

4 m²

Area to paint:

54 − 2 − 4

= 48 m²

Therefore:

48 m²

of wall surface needs paint.

Finding a Missing Dimension

Surface area can also be used to determine missing dimensions.

Suppose a rectangular prism has:

SA = 94 cm²

l = 5 cm

w = 3 cm

Find its height.

Use:

SA = 2lw + 2lh + 2wh

Substitute:

94 = 2(5 × 3) + 2(5 × h) + 2(3 × h)

94 = 30 + 10h + 6h

94 = 30 + 16h

Subtract 30:

64 = 16h

Therefore:

h = 4 cm

Finding the Side Length of a Cube

Suppose a cube has surface area:

216 cm²

Use:

SA = 6s²

Substitute:

216 = 6s²

Divide by 6:

36 = s²

Therefore:

s = 6 cm

Comparing Surface Areas

Consider two boxes.

Box A

10 cm × 5 cm × 4 cm

Surface area:

SA = 2[(10 × 5) + (10 × 4) + (5 × 4)]

SA = 2(50 + 40 + 20)

SA = 220 cm²

Box B

8 cm × 6 cm × 5 cm

Surface area:

SA = 2[(8 × 6) + (8 × 5) + (6 × 5)]

SA = 2(48 + 40 + 30)

SA = 236 cm²

Box B requires more material to cover its outside surfaces.

Same Volume, Different Surface Area

Objects can have the same volume but different surface areas.

Consider:

Prism A

1 cm × 4 cm × 8 cm

Volume:

V = 32 cm³

Surface area:

SA = 2(4 + 8 + 32)

SA = 88 cm²

Prism B

2 cm × 4 cm × 4 cm

Volume:

V = 32 cm³

Surface area:

SA = 2(8 + 8 + 16)

SA = 64 cm²

Both have the same volume:

32 cm³

but Prism B has a smaller surface area.

This is important in packaging and engineering because shape can affect how much material is required.

Surface Area and Packaging Design

Packaging designers often want to:

• Protect the product.
• Use as little material as practical.
• Reduce manufacturing costs.
• Reduce packaging waste.
• Make packages easy to stack and transport.

Two packages can hold the same volume but require different amounts of material.

Surface-area calculations help engineers compare these designs.

 

 

 

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Surface Area and Heat Transfer

Surface area also matters in science.

Objects exchange thermal energy through their surfaces.

An object with a larger surface area exposed to its surroundings may transfer thermal energy more rapidly under otherwise similar conditions.

This helps explain why:

• Small pieces of food can cool faster than one large piece.
• Radiators use large surface areas.
• Cooling fins are used on some machines.
• Animals living in different climates may benefit from different body shapes.

Surface area therefore connects mathematics with physics, biology, and engineering.

Surface-Area-to-Volume Ratio

As objects change size, surface area and volume do not increase at the same rate.

Consider a cube.

Cube with side 1 cm

Surface area:

SA = 6(1²) = 6 cm²

Volume:

V = 1³ = 1 cm³

Surface-area-to-volume ratio:

6 : 1

Cube with side 2 cm

Surface area:

SA = 6(2²) = 24 cm²

Volume:

V = 2³ = 8 cm³

Ratio:

24 : 8

Simplify:

3 : 1

Cube with side 3 cm

Surface area:

SA = 6(3²) = 54 cm²

Volume:

V = 3³ = 27 cm³

Ratio:

54 : 27

Simplify:

2 : 1

As the cube becomes larger, its surface-area-to-volume ratio becomes smaller.

This idea becomes very important in biology and physics.

Scaling Surface Area

Suppose every dimension of a cube is doubled.

Original side:

2 cm

Original surface area:

SA = 6(2²)

SA = 24 cm²

Double the side:

4 cm

New surface area:

SA = 6(4²)

SA = 96 cm²

Compare:

96 ÷ 24 = 4

Doubling every dimension increases the surface area by a factor of:

4

because:

2² = 4

Surface Area and Volume Scale Differently

If every dimension is doubled:

Surface area × 4

but:

Volume × 8

This is because:

surface area depends on length²

while:

volume depends on length³

This difference is important in:

• Biology.
• Engineering.
• Architecture.
• Packaging.
• Heat transfer.

Units Must Match

Before calculating surface area, make sure all dimensions use compatible units.

Suppose a box measures:

2 m × 50 cm × 40 cm

Convert to metres:

50 cm = 0.5 m

40 cm = 0.4 m

Now:

SA = 2[(2 × 0.5) + (2 × 0.4) + (0.5 × 0.4)]

SA = 2(1 + 0.8 + 0.2)

SA = 4 m²

Converting Square Units

Remember:

1 m = 100 cm

but:

1 m² = 10,000 cm²

Surface area uses square units, so area-conversion rules apply.

For example:

2 m² = 20,000 cm²

A Reliable Problem-Solving Method

When calculating surface area:

Step 1: Identify the three-dimensional object.

Step 2: Identify all faces that need to be included.

Step 3: Check whether any faces are missing or uncovered.

Step 4: Make sure all dimensions use compatible units.

Step 5: Calculate the area of each type of face.

Step 6: Account for repeated faces.

Step 7: Add the areas.

Step 8: Write the answer using square units.

Step 9: Check whether the answer is reasonable.

Common Mistakes

Using Cubic Units

Incorrect:

SA = 150 cm³

Correct:

SA = 150 cm²

Surface area is an area, not a volume.

Calculating Volume Instead

For a rectangular prism:

l × w × h

calculates volume.

Surface area requires the areas of the faces.

Forgetting Opposite Faces

A rectangular prism has:

2(lw)

2(lh)

2(wh)

Forgetting the second copy of a face gives an incorrect answer.

Counting a Face Twice

When using a net, mark each face after calculating it so that you do not accidentally count it twice.

Including a Face That Does Not Exist

An open-top box has no top.

Do not automatically use the complete rectangular-prism formula when the object is open.

Forgetting to Subtract Openings

Doors, windows, holes, and other uncovered sections may need to be subtracted in practical problems.

Mixing Units

Convert measurements to compatible units before multiplying.

Confusing Surface Area and Volume

Remember:

Surface area = covering

Volume = filling

Check Your Understanding

1. Define surface area.

2. What is a face of a three-dimensional object?

3. How many faces does a rectangular prism have?

4. How many faces does a cube have?

5. What is a net?

6. Why are nets useful when calculating surface area?

7. Write the surface-area formula for a rectangular prism.

8. Write the surface-area formula for a cube.

9. Find the surface area of a rectangular prism measuring 8 cm × 5 cm × 3 cm.

10. Find the surface area of a box measuring 12 cm × 6 cm × 4 cm.

11. Find the surface area of a cube with side length 7 cm.

12. A cube has a surface area of 294 cm². Find its side length.

13. A rectangular prism measures 10 m × 4 m × 3 m. Find its surface area.

14. An open-top box measures 10 cm × 8 cm × 5 cm. Find the area of material required to make it.

15. Explain why the surface area of the box in Question 14 is different from that of a closed box with the same dimensions.

16. A gift box measures 25 cm × 15 cm × 10 cm. Calculate the minimum area of wrapping paper required to cover it completely.

17. Explain why the actual amount of wrapping paper used would probably be greater than your answer to Question 16.

18. A room measures 6 m × 4 m × 3 m. Calculate the total area of its four walls.

19. The room in Question 18 has a door with area 2 m² and windows with a combined area of 5 m². Find the wall area that needs painting.

20. A rectangular aquarium measuring 70 cm × 30 cm × 40 cm has no top. Find the total area of glass required.

21. Explain the difference between surface area and volume.

22. A cube has side length 4 cm. Calculate both its surface area and volume.

23. If every dimension of a cube is doubled, by what factor does its surface area increase?

24. If every dimension of the same cube is doubled, by what factor does its volume increase?

25. Explain why surface area is important in packaging design.

26. Give two examples in which surface area affects heat transfer.

27. A rectangular prism has a surface area of 94 cm², length 5 cm, and width 3 cm. Find its height.

28. Two boxes have the same volume. Must they have the same surface area? Explain.

29. Explain why a compact box may require less packaging material than a long, thin box with the same volume.

30. Give three real-world situations where calculating surface area would be useful.

Key Terms

• Surface area – total area of all the outside surfaces of a three-dimensional object.
• Face – flat surface of a three-dimensional object.
• Edge – line where two faces meet.
• Vertex – point where edges meet.
• Rectangular prism – three-dimensional object with six rectangular faces.
• Cube – rectangular prism with six identical square faces.
• Net – two-dimensional arrangement of faces that can be folded into a three-dimensional object.
• Square unit – unit used to measure area, such as cm² or m².
• Volume – amount of three-dimensional space inside an object.
• Surface-area-to-volume ratio – comparison between an object's surface area and its volume.

Key Takeaways

• Surface area is the total area of all the outside faces of a three-dimensional object.
• Surface area is measured in square units, such as cm² and m².
• A rectangular prism has six faces arranged in three identical pairs.
• For a rectangular prism:

SA = 2lw + 2lh + 2wh

• This can also be written:

SA = 2(lw + lh + wh)

• A cube has six identical square faces.
• For a cube:

SA = 6s²

• A net shows the faces of a three-dimensional object laid flat.
• Nets can help identify which faces need to be included.
• Open containers may have fewer surfaces than closed objects.
• Doors, windows, and other openings may need to be subtracted.
• Surface area measures the amount needed to cover an object.
• Volume measures the amount of space available to fill an object.
• Packaging, painting, construction, manufacturing, and heat transfer all involve surface area.
• Two objects with the same volume can have different surface areas.
• Doubling every dimension multiplies surface area by 4 but volume by 8.
• Surface-area-to-volume ratio decreases as similarly shaped objects become larger.

A useful way to remember the difference is:

Surface area = covering the outside.

Volume = filling the inside.