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Year 6 Maths Length, Perimeter & Area

Area of composite figures

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Theory

A composite figure is made of rectangles joined together. Find its area by splitting it into rectangles and adding their areas, or by finding the surrounding rectangle and subtracting the part that is missing. This Year 6 maths page shows both methods on L, T and U shapes, how to work out sides that are not marked, and how one method checks the other.

A composite figure is a shape made from two or more simple shapes. A rectilinear figure has only straight sides meeting at right angles.

Split and add: draw a line to cut the figure into rectangles, find each area, then add.

Surround and subtract: find the area of the smallest rectangle that encloses the figure, then subtract each missing piece.

Unmarked sides: opposite sides of the surrounding rectangle are equal, so a missing side is a total minus a part, such as \(10 - 4 = 6\) m.

L-shape split into two rectangles An L-shape 12 cm along the bottom and 9 cm up the left side, split by a dashed line into a bottom rectangle 12 cm by 3 cm (area 36) and a top rectangle 5 cm by 6 cm (area 30). 12 cm 3 cm 7 cm 6 cm 5 cm 9 cm 36 30
Split: \(12 \times 3 + 5 \times 6 = 36 + 30 = 66\,\text{cm}^2\).
L-shape inside its surrounding rectangle The same L-shape inside a dashed 12 cm by 9 cm rectangle. The missing top-right piece is 7 cm by 6 cm. 12 cm 3 cm 7 cm 6 cm 5 cm 9 cm 42
Subtract: \(12 \times 9 - 7 \times 6 = 108 - 42 = 66\,\text{cm}^2\).
L-shape on a centimetre grid An L-shape on a grid of 1 cm squares: a bottom strip 5 squares by 2 and a top part 2 squares by 2, 14 squares in all.
\(5 \times 2 + 2 \times 2 = 14\) squares \(= 14\,\text{cm}^2\).

Two methods for the L-shape in the diagrams:

\[ A = 12 \times 3 + 5 \times 6 = 66\,\text{cm}^2 \]
A =12×3+5×6=66 cm
\[ A = 12 \times 9 - 7 \times 6 = 66\,\text{cm}^2 \]
A =12×9−7×6=66 cm
Find all the sides first. Each rectangle needs a length and a width; mark any missing sides on the diagram.
Check with the other method. Split-and-add and surround-and-subtract must give the same area.

Split and add

  1. Draw one line to make two rectangles (no overlap).
  2. Find the length and width of each.
  3. Multiply for each area and add.

Surround and subtract

  1. Multiply the full width by the full height.
  2. Subtract the area of each missing piece.
Example 1 — Split and add
Use the first diagram. Find the area of the L-shape.
Solution
\(12 \times 3\)\(=\)\(36\)
\(5 \times 6\)\(=\)\(30\)
\(36 + 30\)\(=\)\(66\)
36+30=66 cm

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

Example 2 — Surround and subtract
Use the second diagram to check the area.
Solution
\(12 \times 9\)\(=\)\(108\)
\(7 \times 6\)\(=\)\(42\)
\(108 - 42\)\(=\)\(66\)
108−42=66 cm

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

Example 3 — On a grid
Use the grid diagram. Each square is \(1\,\text{cm}^2\). Find the area.
Solution
\(5 \times 2 + 2 \times 2\)\(=\)\(14\)
10+4=14 cm

\(14\,\text{cm}^2\).

Example 4 — T-shape
Find the area of this T-shape.
T-shape A T-shape: a top bar 10 cm wide and 2 cm tall, and a stem 4 cm wide and 5 cm tall under its middle. 10 cm 2 cm 5 cm 4 cm 5 cm
Solution
\(10 \times 2\)\(=\)\(20\)
\(4 \times 5\)\(=\)\(20\)
\(20 + 20\)\(=\)\(40\)
20+20=40 cm

\(40\,\text{cm}^2\).

Example 5 — Unmarked sides
An L-shaped courtyard is \(16\) m along the bottom and \(10\) m up the left. The top is \(6\) m and the short right side is \(4\) m.
(i) Find the two unmarked sides.
(ii) Find the area.
L-shaped courtyard An L-shaped courtyard: 16 m along the bottom, 4 m up the right, an unmarked step across, an unmarked step up, 6 m across the top and 10 m down the left. 16 m 4 m ? ? 6 m 10 m
Solution
\(10 - 4\)\(=\)\(6\)
\(16 - 6\)\(=\)\(10\)
\(16 \times 4 + 6 \times 6\)\(=\)\(100\)
64+36=100 m

(i) The step up is \(6\) m and the step across is \(10\) m.
(ii) \(100\,\text{m}^2\)

Example 6 — Check by subtracting
Check the courtyard area with the surrounding rectangle.
Solution
\(16 \times 10\)\(=\)\(160\)
\(10 \times 6\)\(=\)\(60\)
\(160 - 60\)\(=\)\(100\)
160−60=100 m

\(100\,\text{m}^2\), the same.

Example 7 — U-shape
A U-shape fits in a \(10\) cm by \(6\) cm rectangle with a \(4\) cm by \(4\) cm notch cut from the top. Find its area.
U-shape A U-shape 10 cm wide and 6 cm tall with a notch 4 cm wide and 4 cm deep cut from the middle of the top. 10 cm 6 cm 4 cm 4 cm 6 cm
Solution
\(10 \times 6\)\(=\)\(60\)
\(4 \times 4\)\(=\)\(16\)
\(60 - 16\)\(=\)\(44\)
60−16=44

\(44\,\text{cm}^2\).

Example 8 — Find the mistake
For the courtyard, Max multiplied the two longest sides, \(16 \times 10 = 160\), and wrote \(160\,\text{m}^2\). What is wrong?
Solution

That is the surrounding rectangle. The missing \(10 \times 6 = 60\,\text{m}^2\) corner must be subtracted: \(100\,\text{m}^2\).

Common pitfalls

Multiplying the two longest sides. That includes the missing corner.
Using a side that belongs to the other rectangle. Each part has its own length and width.
Overlapping parts. Split with one line so no square is counted twice.
Forgetting square units. Area is in \(\text{cm}^2\) or \(\text{m}^2\).

Frequently asked questions

How do you find the area of a composite figure?

Split it into rectangles and add, or subtract the missing parts from the surrounding rectangle.

How do you find a side that is not marked?

Use the opposite sides: \(10 - 4 = 6\) m.

Does it matter where I split the shape?

No: every correct split gives the same total.

How can I check my answer?

Use the other method; the areas must match.

What is the area of a 10 cm by 6 cm rectangle with a 4 cm by 4 cm notch?

\(60 - 16 = 44\,\text{cm}^2\).

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