Perimeter/area of composite shapes | Year 8

01 / LEARN

Understand the idea.

Make a useful plan

For a composite area, choose familiar regions that cover the shape exactly once. Add regions that join without overlap, or subtract a missing region from a larger shape. A sketch helps show which areas belong in the calculation.

In the L-shaped diagram, the enclosing rectangle has area $8 \times 6 = 48 {cm}^{2}$ . The removed square has area $3 \times 3 = 9 {cm}^{2}$ . Subtracting leaves $39 {cm}^{2}$ ; the construction lines do not change the region.

Trace the perimeter

For perimeter, trace the exposed boundary once, including any inward corners. Lines drawn just to divide the area are internal and are not counted. Label missing boundary lengths from the given dimensions before adding.

A semicircle has a curved edge and a straight diameter. Its arc is half a full circumference, $π r$ , and its diameter is $2 r$ . Therefore its full perimeter is $π r + 2 r$ , while its area is $\frac{1}{2} π r^{2}$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

A 12 cm by 8 cm rectangle with a 4 cm by 3 cm rectangle removed from its upper-right corner. The remaining region is lightly shaded and the missing corner has dashed edges.

Worked solution

1
Find the area before the corner is removed.
$A_{whole} = 12 \times 8 = 96 {cm}^{2}$
A rectangle's area is length multiplied by width.
2
Find the area of the cut-out.
$A_{removed} = 4 \times 3 = 12 {cm}^{2}$
Only the smaller rectangular region is removed.
3
Subtract the missing area.
$A_{remaining} = 96 - 12 = 84 {cm}^{2}$
The remaining region and the cut-out together make the original rectangle.

Answer

84 cm².

EXAMPLE 02

A straight edge matters

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

A semicircle with radius 4 cm. Its straight diameter is part of the boundary. The radius is drawn from the centre to the curved edge.

Worked solution

1
Find half the circle's area.
$A = \frac{1}{2} π {(4)}^{2} = 8 π {cm}^{2}$
A semicircle covers half of the corresponding full circle.
2
Find the curved boundary.
$Arc length = \frac{1}{2} (2 π \times 4) = 4 π cm$
The curved edge is half the full circumference.
3
Find the straight boundary.
$d = 2 \times 4 = 8 cm$
The straight edge is the diameter.
4
Add both boundary parts.
$P = (4 π + 8) cm$
The full perimeter includes the arc and the straight edge.

Answer

Area $8 π$ cm²; perimeter $(4 π + 8)$ cm.

03 / YOUR TURN

Try it yourself.

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 & reasoning

30 cm².

The rectangles do not overlap, so their areas add: $18 + 12 = 30 {cm}^{2}$ . Joining along a line creates no overlapping area.

Keep practising

6 worksheet questions · 16 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Make a useful plan

Trace the perimeter

See the method in action.

Subtract a missing corner

Find the area before the corner is removed.

Find the area of the cut-out.

Subtract the missing area.

A straight edge matters

Find half the circle's area.

Find the curved boundary.

Find the straight boundary.

Add both boundary parts.

Try it yourself.

Check your understanding

Keep practising