Find perimeter, area, volume, and surface area of composite shapes

Ontario Grade 8 Mathematics

Spatial Sense

Break a larger shape into familiar parts, then combine the measurements carefully.

A composite shape is made from two or more simpler shapes. Think of a floor plan made from rectangles, or a solid made by joining two boxes. You can find its measurements by dividing it into familiar parts. First decide what is being measured: perimeter is the distance around a flat shape, area is the space inside it, volume is the space inside a solid, and surface area is the total area of a solid’s outside faces. Keep units in mind: perimeter uses units such as centimetres, while area and surface area use square units, and volume uses cubic units.

What you will learn

Find the perimeter of a composite flat shape by adding its outside side lengths.
Find the area of a composite flat shape by splitting it into simpler shapes.
Find the volume of a composite solid by combining the volumes of its parts.
Find the surface area of a composite solid by accounting for faces that are joined inside.

Start with familiar shapes and units

Before working with a composite shape, review a few familiar measurements. A rectangle’s area is its length multiplied by its width. Its perimeter is the sum of its four side lengths. A rectangular prism is a box shape. Its volume is the area of its base multiplied by its height.

A face is one flat surface of a solid. Surface area is the sum of the areas of all the faces on the outside. When two solids are joined, the faces where they touch are inside the new solid. They are not part of its outside surface.

P=sum of outside side lengths

Perimeter measures around a flat shape.
Area measures the inside of a flat shape.
Volume measures space inside a solid.
Surface area measures the outside faces of a solid.

Split, measure, and combine

A useful first move is to look for rectangles or rectangular prisms inside the composite shape. A line you draw to split a shape is not automatically part of its perimeter. For perimeter, count only the boundary that you could trace while staying on the outside.

For area, split a flat shape into parts whose areas you can find. Add the areas if the parts make up the whole shape. If it is easier to imagine a missing corner, find the area of the full rectangle and subtract the missing part.

For volume, split a solid into non-overlapping parts. Find each part’s volume and add them. Make sure the parts fill the solid without leaving gaps or counting the same space twice.

A_{\text{whole}}=A_1+A_2+\cdots

A composite shape can be split into simpler parts.
For perimeter, use outside edges only.
For area and volume, add the parts that make up the shape.

Treat joined faces carefully

Surface area needs an extra check. Imagine painting the outside of a solid. A face covered by another part cannot be painted, so it is not an outside face.

One way to find surface area is to list the exposed faces and add their areas. Another way is to find the surface areas of the separate parts, then subtract the joined faces twice. Each joined face was counted once on each part, but neither face belongs on the outside of the combined solid.

Use square units for area and surface area, and cubic units for volume. For example, a measurement in centimetres gives area in square centimetres and volume in cubic centimetres.

SA_{\text{combined}}=SA_1+SA_2-2A_{\text{joined}}

Count only faces exposed on the outside.
If adding separate surface areas, remove both copies of every joined face.
Write the correct kind of unit with each answer.

Worked example

An L-shaped floor section

A floor section is a rectangle measuring 10 m by 8 m with a 4 m by 3 m rectangle missing from its upper-right corner. Find its area and perimeter.

Find the area
Begin with the full rectangle. Its area is its length multiplied by its width. Subtract the missing corner’s area because that part is not in the floor section.
$10\times8-4\times3=68\text{ m}^2$
Find the outside lengths
The missing corner creates two new outside edges. The remaining top edge is $10-4=6$ m, and the right edge is $8-3=5$ m. Add these and the other outside edges. Do not count the edges around the missing rectangle that do not touch the outside.
$10+5+4+3+6+8=36\text{ m}$

Answer: The area is

68\text{ m}^2

. The perimeter is

36\text{ m}

.

Check: The area is smaller than the full

80\text{ m}^2

rectangle because a corner is missing. The perimeter includes the two new edges around that corner.

Worked example

A solid made from two box-shaped parts

A solid is made by joining two rectangular prisms without overlap. The first measures 6 cm by 4 cm by 2 cm. The second measures 3 cm by 4 cm by 2 cm. Find the total volume.

Find each part’s volume
Multiply length, width, and height for each rectangular prism. The parts do not overlap, so their volumes can be added.
$6\times4\times2+3\times4\times2=72\text{ cm}^3$
State the total
The joined faces affect surface area, but they do not remove any of the solid’s space. The total volume is the sum of both parts.
$48+24=72\text{ cm}^3$

Answer: The total volume is

72\text{ cm}^3

.

Check: The first part has volume

48\text{ cm}^3

, and the second has volume

24\text{ cm}^3

. Their sum is

72\text{ cm}^3

.

Worked example

Surface area of joined prisms

A rectangular prism measuring 5 cm by 4 cm by 2 cm has a smaller prism measuring 2 cm by 4 cm by 3 cm attached to one of its faces. The parts touch across a 2 cm by 4 cm rectangle. Find the surface area of the combined solid.

Find the separate surface areas
A rectangular prism has three pairs of equal faces. For each part, find the areas of the three different face sizes, add them, and double the result.
$2(5\times4+5\times2+4\times2)+2(2\times4+2\times3+4\times3)=128\text{ cm}^2$
Remove the joined faces
The touching rectangle has area $2\times4=8\text{ cm}^2$ . It was counted once on each prism, so subtract two copies. The remaining area is the outside surface of the combined solid.
$128-2(8)=112\text{ cm}^2$

Answer: The surface area is

112\text{ cm}^2

.

Check: Subtracting two copies of the joined face removes both surfaces that became hidden inside the combined solid.

Common mistakes and how to avoid them

Adding every drawn edge when finding perimeter.

Correction: Trace the outside boundary only. A line inside the shape is not part of its perimeter.

Adding the missing corner’s area instead of subtracting it.

Correction: If the shape is a full rectangle with a part removed, subtract the missing part’s area.

Counting a joined face as part of surface area.

Correction: Joined faces are inside. Exclude them, or subtract both counted copies.

Writing square units for volume or cubic units for area.

Correction: Use linear units for perimeter, square units for area and surface area, and cubic units for volume.

Lesson summary

Split composite shapes into familiar parts.
For perimeter, add only the outside side lengths.
For area and volume, combine the parts that make up the whole.
For surface area, count only exposed faces.
Check that the final units match the measurement.

Check your understanding

Question 1

A 9 cm by 6 cm rectangle has a 3 cm by 2 cm corner removed. What is its area?

$48\text{ cm}^2$
$54\text{ cm}^2$
$60\text{ cm}^2$
correctIndex: 0,

Show answer and explanation

48\text{ cm}^2

The full rectangle has area

54\text{ cm}^2

. The missing corner has area

6\text{ cm}^2

, so the remaining area is

48\text{ cm}^2

.

Question 2

Two non-overlapping box-shaped parts have volumes of

30\text{ cm}^3

and

18\text{ cm}^3

. What is the total volume?

$12\text{ cm}^3$
$48\text{ cm}^3$
$540\text{ cm}^3$
correctIndex: 1,

Show answer and explanation

48\text{ cm}^3

Add the volumes because the parts do not overlap:

30+18=48\text{ cm}^3

.

Key terms

Composite shape: A shape made by joining or combining simpler shapes.
Face: One flat surface of a solid.
Exposed face: A face that is on the outside of a solid and is not hidden by another part.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation E2.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E2.3 · Find perimeter, area, volume, and surface area of composite shapes

Ontario Grade 8 Mathematics

Break a larger shape into familiar parts, then combine the measurements carefully.

What you will learn

Start with familiar shapes and units

Split, measure, and combine

Treat joined faces carefully

An L-shaped floor section

A solid made from two box-shaped parts

Surface area of joined prisms

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Grade 8 Math

About this lesson and its review