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Year 5 Maths Length, Perimeter & Area

Area of composite figures

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Theory

To find the area of an L-shape, split it into rectangles, find each area with \(\text{length} \times \text{width}\), and add. Or find the area of the surrounding rectangle and subtract the piece that is cut out. This Year 5 maths page shows both methods, how to find missing sides first, and why you cannot just multiply the two longest sides.

Area is the amount of flat space inside a shape, measured in square units such as \(\text{cm}^2\) or \(\text{m}^2\). A rectangle has area \(\text{length} \times \text{width}\).

Split and add: draw a line to cut the figure into two rectangles. Find each area and add them.

Surround and subtract: find the area of the whole rectangle the figure fits in, then subtract the area of the missing corner.

Every method gives the same answer, so a second method is a good check.

L-shape split into two rectangles An L-shaped figure 10 cm wide at the bottom and 9 cm tall on the left, split by a dashed line into a bottom rectangle 10 cm by 4 cm (area 40) and a top rectangle 6 cm by 5 cm (area 30). 10 cm 4 cm 4 cm 5 cm 6 cm 9 cm 40 30
Split: \(10 \times 4 + 6 \times 5 = 40 + 30 = 70\,\text{cm}^2\).
L-shape inside its surrounding rectangle The same L-shape inside a dashed 10 cm by 9 cm rectangle. The missing top-right corner is 4 cm by 5 cm. 10 cm 4 cm 4 cm 5 cm 6 cm 9 cm 20
Subtract: \(10 \times 9 - 4 \times 5 = 90 - 20 = 70\,\text{cm}^2\).
L-shape on a centimetre grid An L-shape on a grid of 1 cm squares: a bottom strip 6 squares by 2 squares and a top part 3 squares by 3 squares, 21 squares in all.
On a grid: \(6 \times 2 + 3 \times 3 = 12 + 9 = 21\) squares \(= 21\,\text{cm}^2\).

Two ways to find the area of an L-shape:

\[ A = 10 \times 4 + 6 \times 5 = 70\,\text{cm}^2 \]
A =10×4+6×5=70 cm
\[ A = 10 \times 9 - 4 \times 5 = 70\,\text{cm}^2 \]
A =10×9−4×5=70 cm
Find missing sides first. Each rectangle needs both its length and its width.
Square units. Area is written in \(\text{cm}^2\) or \(\text{m}^2\), not cm or m.

Split and add

  1. Draw a line to split the figure into two rectangles.
  2. Find the length and width of each (work out missing sides).
  3. Multiply to get each area, then add.

Surround and subtract

  1. Find the area of the surrounding rectangle.
  2. Find the area of the cut-out corner.
  3. Subtract.
Example 1 — Split and add
Use the first diagram. Find the area of the L-shape.
Solution
\(10 \times 4\)\(=\)\(40\)
\(6 \times 5\)\(=\)\(30\)
\(40 + 30\)\(=\)\(70\,\text{cm}^2\)
10×4+6×5=70 cm

\(70\,\text{cm}^2\).

Example 2 — Surround and subtract
Use the second diagram. Find the area by subtracting.
Solution
\(10 \times 9\)\(=\)\(90\)
\(4 \times 5\)\(=\)\(20\)
\(90 - 20\)\(=\)\(70\,\text{cm}^2\)
90−20=70 cm

\(70\,\text{cm}^2\), the same answer.

Example 3 — Count on a grid
Use the grid diagram. Each square is \(1\,\text{cm}^2\). Find the area.
Solution
\(6 \times 2 + 3 \times 3\)\(=\)\(21\,\text{cm}^2\)
6×2+3×3=21 cm

\(21\,\text{cm}^2\).

Example 4 — T-shape
A T-shape has a top bar \(14\) cm by \(3\) cm and a stem \(4\) cm wide and \(6\) cm tall. Find its area.
T-shape A T-shape: a top bar 14 cm wide and 3 cm tall, and a stem 4 cm wide and 6 cm tall below its middle. 14 cm 3 cm 6 cm 4 cm
Solution
\(14 \times 3\)\(=\)\(42\)
\(4 \times 6\)\(=\)\(24\)
\(42 + 24\)\(=\)\(66\,\text{cm}^2\)
14×3+4×6=66 cm

\(66\,\text{cm}^2\).

Example 5 — Missing sides first
An L-shaped garden bed is \(14\) m along the bottom and \(7\) m up the left side. The top is \(9\) m and the right side is \(3\) m. Find its area.
L-shaped garden bed An L-shaped garden bed: 14 m along the bottom, 3 m up the right, an unmarked step across, an unmarked step up, 9 m across the top and 7 m down the left. 14 m 3 m ? 9 m 7 m
Solution
\(7 - 3\)\(=\)\(4\text{ m}\)
\(14 \times 3 + 9 \times 4\)\(=\)\(42 + 36\)
\(42 + 36\)\(=\)\(78\,\text{m}^2\)
14×3+9×4=78 m

The step is \(4\) m, and the area is \(78\,\text{m}^2\).

Example 6 — Check by subtracting
Check the garden bed area using the surrounding \(14\) m by \(7\) m rectangle.
Solution
\(14 \times 7 - 5 \times 4\)\(=\)\(98 - 20\)
\(98 - 20\)\(=\)\(78\,\text{m}^2\)
98−20=78 m

The cut-out is \(5\) m by \(4\) m; the area is \(78\,\text{m}^2\) again.

Example 7 — Find the mistake
For the garden bed, Rosa multiplied the two longest sides, \(14 \times 7 = 98\), and wrote \(98\,\text{m}^2\). What is wrong?
Solution

That is the area of the whole surrounding rectangle. She must subtract the missing \(5 \times 4 = 20\,\text{m}^2\) corner: \(78\,\text{m}^2\).

Common pitfalls

Multiplying the two longest sides. That gives the surrounding rectangle, which includes the missing corner.
Using the wrong width for a part. Each rectangle has its own length and width; read them carefully from the diagram.
Counting the overlap twice. Split with one line so the two rectangles do not overlap.
Writing cm instead of cm². Area is in square units, such as \(\text{cm}^2\).

Frequently asked questions

How do you find the area of an L-shape?

Split it into two rectangles, find each area and add: \(10 \times 4 + 6 \times 5 = 70\,\text{cm}^2\).

Is there another way?

Find the surrounding rectangle and subtract the missing corner: \(10 \times 9 - 4 \times 5 = 70\,\text{cm}^2\).

Does it matter where you split the shape?

No. Any split into rectangles gives the same total area.

What if a side is not marked?

Work it out from the parallel sides first: \(7 - 3 = 4\) m.

What unit is area measured in?

Square units, such as \(\text{cm}^2\) or \(\text{m}^2\).

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