Year 9 Maths Core Surface Area and Volume

Perimeters and Areas of Composite Shapes

20 practice questions 1 video lesson Theory + worked examples

Composite shapes are made of simpler shapes joined together or cut away. This NSW Year 9 Core maths subtopic revises perimeter and area from Stage 4 (NESA outcome MA4‑ARE‑C‑01), with semicircles attached, paths around pools and missing sides to work out first.

You will find perimeters by working around the outside, find a missing side from the sides opposite it, and find areas either by splitting a shape into rectangles and triangles or by enclosing it and subtracting the missing piece. You will also add a semicircle, as in a garden bed with a curved end, find the area of a path between two rectangles and its paving cost, and work backwards from the lawn left after a flower bed is dug out to find the lawn's width. It is part of Year 9 Core Surface Area and Volume.

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Perimeters and Areas of Composite Shapes - Video - area perimeter composite figures Watch

Theory

A composite shape is made of simpler shapes joined together. This Year 9 maths page revises how to find its perimeter, including missing sides, and its area by adding pieces or subtracting a hole. It also covers semicircles and paths around a shape.

A composite shape is built from shapes you already know: rectangles, triangles, parallelograms, trapeziums and parts of circles.

The perimeter is the distance around the outside edge. Walk round the shape once and add every side. Lines drawn inside the shape do not count.

On a shape with right-angled corners, a missing side comes from the sides opposite it. The pieces going one way add up to the full length on the other side.

The area can be found two ways. Add: cut the shape into pieces and add their areas. Subtract: enclose it in a bigger shape and take away the missing part. Choose the way that needs fewer missing lengths.

Add: cut the shape into pieces you know, find each area, and add them.

Subtract: complete the shape to a rectangle, then take away the missing piece (red). Both methods give the same area.

The shapes you build from:

\[A_{\text{rectangle}} = lb \qquad A_{\text{triangle}} = \dfrac{1}{2}bh \qquad A_{\text{parallelogram}} = bh\]

A_{triangle} = \frac{1}{2} b h

\[A_{\text{trapezium}} = \dfrac{h}{2}(a + b) \qquad A_{\text{circle}} = \pi r^2\]

A_{trapezium} = \frac{h}{2} (a + b)

A semicircle is half a circle:

\[A = \dfrac{1}{2}\pi r^2 \qquad \text{curved edge} = \dfrac{1}{2} \times 2\pi r = \pi r\]

A = \frac{1}{2} π r^{2}

A slanted side. If a side is slanted, draw a right-angled triangle under it and use Pythagoras. A trapezium with parallel sides \(9\) and \(13\) and height \(3\) has a slanted side of \(\sqrt{4^2 + 3^2} = 5\).

Working backwards. If the area is given, write it with the unknown and solve. A rectangle \(9\) m long with a \(2\) m by \(2\) m square cut out leaves \(50 \text{ m}^2\): \(9w - 4 = 50\), so \(w = 6\) m.

Equal marks. Sides with the same tick marks are equal in length. Use them to fill in unlabelled sides.

How to find the perimeter and area of a composite shape

Find any missing sides from the sides opposite them, or from matching tick marks.
Perimeter: walk round the outside once and add every side. A curved edge is part of a circumference.
Area: split the shape into pieces and add, or enclose it and subtract the missing part.
Use the right formula for each piece, then add or subtract.
Check the units: perimeter in m or cm, area in $\text{m}^2$ or $\text{cm}^2$.

Example 1 — Missing sides, then the perimeter

An L-shape has right-angled corners. Its bottom is $14$ m and its left side is $9$ m. A step $6$ m wide and $4$ m deep is missing from the top right. Find the perimeter.

Solution

Find the two missing sides first. The two top pieces must add to the $14$ m bottom. The two right pieces must add to the $9$ m left side.

$\text{top}$	$=$	$14 - 6 = 8 \text{ m}$
$\text{right}$	$=$	$9 - 4 = 5 \text{ m}$

Now walk round the outside once.

$P$	$=$	$14 + 5 + 6 + 4 + 8 + 9$
	$=$	$\mathbf{46} \text{ m}$

P = 14 + 5 + 6 + 4 + 8 + 9 = 46

Example 2 — The area, two ways

Find the area of the L-shape in Example 1.

Solution

Add: cut it with a line into two rectangles.

$A$	$=$	$14 \times 5 + 8 \times 4$
	$=$	$70 + 32 = \mathbf{102} \text{ m}^2$

Subtract: enclose it in a $14$ by $9$ rectangle and take away the $6$ by $4$ corner.

$A$	$=$	$14 \times 9 - 6 \times 4$
	$=$	$126 - 24 = \mathbf{102} \text{ m}^2$

A = 14 \times 9 - 6 \times 4 = 102

The cut line helps with the area only. It is not part of the perimeter.

Example 3 — A semicircle on a rectangle

A rectangle $10$ cm wide and $6$ cm high has a semicircle on top, its diameter the full width. Find
(i) the area
(ii) the perimeter, each to two decimal places.

Solution

(i) The diameter is $10$ cm, so the radius is $5$ cm. Add the rectangle and half a circle.

$A$	$=$	$10 \times 6 + \dfrac{1}{2} \times \pi \times 5^2$
	$=$	$60 + 39.269\ldots$
	$\approx$	$\mathbf{99.27} \text{ cm}^2$

(ii) The top edge is now curved: half a circumference, $\dfrac{1}{2} \times 2\pi r = \pi r$. The other three sides are straight.

$P$	$=$	$10 + 6 + 6 + \pi \times 5$
	$=$	$22 + 15.707\ldots$
	$\approx$	$\mathbf{37.71} \text{ cm}$

P = 22 + 5 π \approx 37.71

Example 4 — A path around a pond

A pond $6$ m by $3$ m has a path $1$ m wide all the way round. Paving costs $30 per square metre. Find the area of the path and the cost.

Solution

The path adds $1$ m on both sides, so the outer rectangle is $8$ m by $5$ m. The path is the outer rectangle minus the pond.

$\text{outer}$	$=$	$8 \times 5 = 40 \text{ m}^2$
$\text{pond}$	$=$	$6 \times 3 = 18 \text{ m}^2$
$\text{path}$	$=$	$40 - 18 = \mathbf{22} \text{ m}^2$

$\text{cost}$

$=$

$22 \times \$30 = \mathbf{\$660}$

(8 \times 5 - 6 \times 3) \times 30 = 660

The path costs $660.

Common pitfalls

A cut line is not part of the perimeter. It only helps with the area. The perimeter follows the outside edge.

Find the missing sides first. A shape can have two unmarked sides. Both come from the marked sides opposite them.

A path goes on both sides. A path $2$ m wide around a $10$ m pool makes the outer length $14$ m, not $12$ m.

Use the radius in the circle formula. If the diameter is $10$, the radius is $5$. A semicircle is half of $\pi r^2$.

Frequently asked questions

What is a composite shape?

A composite shape is made of two or more simple shapes joined together, such as a rectangle with a triangle or a semicircle on top.

How do you find the area of a composite shape?

Either split it into simple shapes, find each area and add them, or enclose it in a bigger shape and subtract the part that is missing. Both give the same answer.

How do you find a missing side on a composite shape?

On a shape with right-angled corners, the sides going one way add up to the full side opposite them. If the bottom is $14$ m and one top piece is $6$ m, the other is $8$ m.

How do you find the perimeter of a shape with a semicircle?

Add the straight sides, then add the curved edge. The curved edge of a semicircle is half the circumference: $\pi r$.

How do you find the area of a path around a rectangle?

Find the area of the outer rectangle, including the path on both sides, then subtract the area of the inner rectangle.

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Areas of Quadrilaterals

\(\text{top}\)	\(=\)	\(14 - 6 = 8 \text{ m}\)
\(\text{right}\)	\(=\)	\(9 - 4 = 5 \text{ m}\)

\(P\)	\(=\)	\(14 + 5 + 6 + 4 + 8 + 9\)
\(\)	\(=\)	\(\mathbf{46} \text{ m}\)

\(A\)	\(=\)	\(14 \times 5 + 8 \times 4\)
\(\)	\(=\)	\(70 + 32 = \mathbf{102} \text{ m}^2\)

\(A\)	\(=\)	\(14 \times 9 - 6 \times 4\)
\(\)	\(=\)	\(126 - 24 = \mathbf{102} \text{ m}^2\)

\(A\)	\(=\)	\(10 \times 6 + \dfrac{1}{2} \times \pi \times 5^2\)
\(\)	\(=\)	\(60 + 39.269\ldots\)
\(\)	\(\approx\)	\(\mathbf{99.27} \text{ cm}^2\)

\(P\)	\(=\)	\(10 + 6 + 6 + \pi \times 5\)
\(\)	\(=\)	\(22 + 15.707\ldots\)
\(\)	\(\approx\)	\(\mathbf{37.71} \text{ cm}\)

\(\text{outer}\)	\(=\)	\(8 \times 5 = 40 \text{ m}^2\)
\(\text{pond}\)	\(=\)	\(6 \times 3 = 18 \text{ m}^2\)
\(\text{path}\)	\(=\)	\(40 - 18 = \mathbf{22} \text{ m}^2\)

Perimeters and Areas of Composite Shapes

📖 Theory

How to find the perimeter and area of a composite shape

Common pitfalls

Frequently asked questions

Theory