Area of Composite Shapes
A composite shape is made from rectangles, triangles and parallelograms joined together, or a large shape with a piece removed. In NSW Year 7 maths (NESA outcome MA4‑ARE‑C‑01) you find its area by splitting it into simple parts.
You will add the parts or subtract a cut-out, work out missing lengths on L-shaped and stepped figures first, check with a second way of splitting, and find a missing length when the total area is known. Lawns with ponds, paths around gardens and shed walls with doors are typical examples. It is part of Year 7 Area and Volume.
Every question with a fully worked solution.
- Area of Composite Shapes - Video - Area of Compound Shapes Watch
Theory
A composite shape is made of two or more simple shapes joined together. Its area is the total of the parts (split and add), or a big shape minus the part that is missing (take away). This Year 7 maths page shows both methods, how to find missing lengths and how to check one method with another.
Method 1: add the parts. Split the shape with a dashed line into simple shapes. Find each area on its own line, then add: \(A = A_1 + A_2\).
Method 2: take away. Find the area of the whole big shape, then subtract the missing part (a cut-out, a pond, a door): \(A = A_1 - A_2\).
Find missing lengths first. In a shape with all right angles, opposite sides match: in the figure in Diagrams, bottom \(=\) top \(+\) step, so \(10 = 6 + 4\), and left \(=\) right \(+\) step, so \(8 = 5 + 3\).
Use the right formula for each part: rectangle \(A = l \times w\); square \(A = s \times s\); triangle \(A = \dfrac{1}{2} \times b \times h\); parallelogram \(A = b \times h\).
Know the total area? Work backwards: subtract the part you can find, then divide what is left by the side you know.
Two methods:
How to find the area of a composite shape
- Find any missing lengths.
- Split into simple shapes (or see the big shape minus a missing part).
- Find each area with its own formula.
- Add (or subtract), then check with another method.
| \(4 \times 5\) | \(=\) | \(20\) |
| \(\dfrac{1}{2} \times 6 \times 5\) | \(=\) | \(15\) |
| \(20 + 15\) | \(=\) | \(35\) |
Area \(= 35\) square units.
| \(6 \times 6\) | \(=\) | \(36\) |
| \(\dfrac{1}{2} \times 6 \times (6 - 3)\) | \(=\) | \(9\) |
| \(36 - 9\) | \(=\) | \(27\) |
The triangle has height \(6 - 3 = 3\). Area \(= 27\) square units.
| \(11 - 5\) | \(=\) | \(6\) |
| \(5 \times 8\) | \(=\) | \(40\) |
| \(6 \times 4\) | \(=\) | \(24\) |
| \(40 + 24\) | \(=\) | \(64\) |
\(64~\text{cm}^2\). Check: \(11 \times 8 - 6 \times 4 = 88 - 24 = 64\).
| \(30 \times 24\) | \(=\) | \(720\) |
| \(20 \times 14\) | \(=\) | \(280\) |
| \(720 - 280\) | \(=\) | \(440\) |
\(440~\text{cm}^2\)
| \(12 \times 8\) | \(=\) | \(96\) |
| \(\dfrac{1}{2} \times 6 \times 4\) | \(=\) | \(12\) |
| \(96 - 12\) | \(=\) | \(84\) |
| \(84 \times 15\) | \(=\) | \(1260\) |
Lawn \(84~\text{m}^2\); the turf costs \(\$1260\).
| \(40 \times 25\) | \(=\) | \(1000\) |
| \(1000 \times (1 + 2 + 3)\) | \(=\) | \(6000\) |
Split into columns of \(1\), \(2\) and \(3\) blocks, each block \(1000~\text{cm}^2\): \(6000~\text{cm}^2\).
| \(45 - 27\) | \(=\) | \(18\) |
| \(18 \div 6\) | \(=\) | \(3\) |
Part B is \(3\) m long.
Common pitfalls
Frequently asked questions
What is a composite shape?
A shape made of two or more simple shapes joined together.
How do you find the area of a composite shape?
Split it into simple shapes and add their areas, or take the missing part away from a bigger shape.
How do you find a missing length?
Opposite sides match: the total length one way equals the total the other way.
Do both methods give the same answer?
Yes. Use one to check the other.
How do you find the area of a frame?
Area of the whole rectangle minus the area of the opening.