Perimeter/area of composite shapes | Year 8

01 / LEARN

The idea, step by step

Make a useful plan

A composite shape can be split into rectangles, triangles or parts of circles. Mark the pieces so they cover the original region exactly once. Alternatively, enclose it in a simple shape and subtract the missing piece.

For the L shape shown, the enclosing rectangle is $8 \times 6$ cm and the missing square is $3 \times 3$ cm. Its area is $48 - 9 = 39$ cm². Internal joining lines are not new outside edges.

Trace the perimeter

Perimeter is found by travelling once around the outside. Include an indentation if it is part of the boundary. Do not count a line used only to divide the area into pieces.

A semicircle’s perimeter includes both its curved edge and its straight diameter. Half of $2 π r$ is only the curved part. A correct formula for the full semicircle boundary is $π r + 2 r$ .

A common trap

Adding every line on the drawing to get the perimeter.

Instead: Trace the outside boundary only. Internal construction lines do not belong to the perimeter.

02 / WORKED EXAMPLES

See the thinking

Example 1

Subtract a missing corner

A rectangle is 12 cm by 8 cm. A 4 cm by 3 cm rectangle is cut from one corner. Find the remaining area.

Whole area: $12 \times 8 = 96$ cm².
Removed area: $4 \times 3 = 12$ cm².
Remaining area: $96 - 12 = 84$ cm².

84 cm².

Example 2

A straight edge matters

A semicircular window has radius 4 cm. Find its area and its full perimeter exactly.

Area is half a circle: $A = \frac{1}{2} π {(4)}^{2} = 8 π$ cm².
The curved boundary is $π r = 4 π$ cm.
The diameter is 8 cm. Add it to the curved boundary.

Area

8 π

cm²; perimeter

(4 π + 8)

cm.

Check your understanding

Two rectangles of areas 18 cm² and 12 cm² join along an edge without overlapping. What is the total area?

Show answer

30 cm².

Add the areas: $18 + 12 = 30$ . Their shared edge has no area.

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A 10 cm by 6 cm rectangle has a 2 cm by 3 cm corner removed. Find the area.
2 marks
Two non-overlapping rectangles have dimensions 8 m by 3 m and 4 m by 5 m. Find their combined area.
2 marks
Find the area of a semicircle of radius 6 cm.
2 marks
Find the full perimeter of a semicircle of radius 3 m.
3 marks
An L shape is made by removing a 3 cm by 2 cm top-right corner from a 9 cm by 5 cm rectangle. Find its area and perimeter.
4 marks
A 6 m by 4 m garden contains a circular bed of radius 1 m. Find the area outside the bed.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A 10 cm by 6 cm rectangle has a 2 cm by 3 cm corner removed. Find the area.
2 marks
$10 (6) - 2 (3) = 60 - 6 = 54$ cm².
54 cm².
Two non-overlapping rectangles have dimensions 8 m by 3 m and 4 m by 5 m. Find their combined area.
2 marks
$8 (3) + 4 (5) = 24 + 20 = 44$ m².
44 m².
Find the area of a semicircle of radius 6 cm.
2 marks
$A = \frac{1}{2} π {(6)}^{2} = 18 π$ cm².
$18 π$ cm².
Find the full perimeter of a semicircle of radius 3 m.
3 marks
Curved length is $π (3) = 3 π$ m.
Straight diameter is 6 m.
Add to obtain $3 π + 6$ m.
$(3 π + 6)$ m.
An L shape is made by removing a 3 cm by 2 cm top-right corner from a 9 cm by 5 cm rectangle. Find its area and perimeter.
4 marks
Area: $9 (5) - 3 (2) = 39$ cm².
The missing outer lengths are replaced by equal inward lengths.
Perimeter remains $2 (9 + 5) = 28$ cm.
Area 39 cm²; perimeter 28 cm.
A 6 m by 4 m garden contains a circular bed of radius 1 m. Find the area outside the bed.
3 marks
Rectangle area: $6 (4) = 24$ m².
Circle area: $π {(1)}^{2} = π$ m².
Subtract the bed area.
$(24 - π)$ m².

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Circumference and area of circles; perimeter/area of composite shapes; choosing and converting area units; cross-sections of prisms; prism volume and capacity; mixed measurement applications.

Official curriculum source ↗

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