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Area of Compound Shapes

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How to find the area of a shape made from rectangles, triangles and circles, by adding the pieces or subtracting a missing piece, and how to use that area to work out bags, tins or costs.

Compound shapes in the exam

A compound shape is made from simpler shapes joined together, such as an L-shape made from two rectangles, or a rectangle with a triangle on top. You find its area from the areas of the simple shapes.

In the exam you are usually given a diagram with some lengths marked and asked to work out the area, for 2 or 3 marks. Some lengths are not marked, so you work them out from the others.

Longer questions, for 4 or 5 marks, use the area in a real problem: how many bags of seed or tins of paint to buy, or how much they cost. Paper 1 has no calculator, so its numbers are kept simple. A shape with a circle in it usually comes on Paper 2 or Paper 3, where you can use a calculator.

Add or subtract

There are two ways to find the area of a compound shape.

  • Add: split the shape into pieces that do not overlap. Find the area of each piece, then add them.
  • Subtract: if the shape is a simple shape with a piece missing, find the area of the whole shape, then subtract the missing piece.

Both ways give the same answer. Choose the one that needs fewer missing lengths.

These are the areas you need:

\text{rectangle}=\text{length}\times\text{width}

\text{triangle}=\frac12\times\text{base}\times\text{height}

\text{circle}=\pi r^2

The height of a triangle is measured at right angles to its base. The radius r of a circle is half its diameter.

Find the missing lengths

In an L-shape every edge goes straight across or straight up. The lengths across the top add up to the length across the bottom. In the same way, the heights on the left match the heights on the right.

So if the whole width is 12 m and one part of it is 7 m, the other part is 12-7=5 m. Only take a width from a width, and a height from a height.

Write each missing length on the diagram before you work out any areas.

Worked example 1: Splitting an L-shape

The diagram shows an L-shape. Work out its area.

  1. Draw a line to split the shape into two rectangles, A and B.
  2. Rectangle B is 10 cm by 3 cm: 10\times3=30\text{ cm}^2

  3. Your turn. The height of rectangle A is not marked. What is it, in centimetres?

    Height of A: 8-3=5\text{ cm}

  4. Rectangle A: 4\times5=20\text{ cm}^2
  5. Answer: 30+20=50\text{ cm}^2

Worked example 2: Subtracting a missing corner

A rectangular floor is 9 m by 8 m. A rectangle 5 m by 3 m is missing from one corner, as the diagram shows. Work out the area of the floor.

  1. The dashed lines complete the whole rectangle. Its area includes the missing corner.
  2. Whole rectangle: 9\times8=72\text{ m}^2
  3. Missing corner: 5\times3=15\text{ m}^2

  4. Your turn. Subtract the missing corner. What is the area of the floor, in square metres?

    72-15=57\text{ m}^2

  5. Answer: 57\text{ m}^2

Triangles and circles in a shape

When a triangle sits on top of a rectangle, the base of the triangle is the top of the rectangle. The height of the triangle goes straight up from that base to the top point.

If you are given the height of the whole shape, subtract the height of the rectangle to find the height of the triangle. Then remember to halve base times height.

For a circle, check whether you are given the radius or the diameter. Halve a diameter to get the radius before you use \pi r^2. Use the \pi key on your calculator, and round only at the end.

Worked example 3: A rectangle with a triangle on top

The diagram shows a sign made from a rectangle and a triangle. Work out the area of the sign.

  1. Find the area of the rectangle and the area of the triangle, then add.
  2. Rectangle: 12\times5=60\text{ cm}^2
  3. The 9 cm is the height of the whole sign. The triangle's height is 9-5=4\text{ cm}

  4. Your turn. The triangle has base 12 cm and height 4 cm. What is its area, in square centimetres?

    Triangle: \frac12\times12\times4=24\text{ cm}^2

  5. Answer: 60+24=84\text{ cm}^2

Worked example 4: Cutting a circle out of a rectangle

A metal plate is a rectangle 14 cm by 10 cm. A circular hole of diameter 8 cm is cut out of it. Work out the area of metal left. Give your answer correct to 1 decimal place.

  1. The metal left is the rectangle subtract the circle.
  2. Rectangle: 14\times10=140\text{ cm}^2
  3. The radius is half the diameter: 8\div2=4\text{ cm}

  4. Your turn. Work out the area of the hole, in square centimetres. Give your answer correct to 2 decimal places.

    Hole: \pi\times4^2=50.265\ldots=50.27\text{ cm}^2 to 2 decimal places.

  5. Metal left: 140-50.265\ldots=89.734\ldots
  6. Answer: 89.7\text{ cm}^2

Borders and paths

A border goes all the way round a shape, like a frame round a picture or a path round a pond. Its area is the outside rectangle subtract the inside rectangle.

The border adds 2 cm on both sides. The outside is 2+9+2=13 cm wide and 2+6+2=10 cm high.

Area of the border: 13\times10-9\times6=130-54=76\text{ cm}^2

Worked example 5: Finding a missing length from the area

A rectangular card is 10 cm by 8 cm. A rectangle 4 cm high is cut from one corner. The area of card left is 68\text{ cm}^2. Work out the width, w, of the piece cut out.

  1. Start with the whole card: 10\times8=80\text{ cm}^2
  2. Area cut out: 80-68=12\text{ cm}^2

  3. Your turn. The piece cut out is a rectangle with area 12\text{ cm}^2 and height 4 cm. What is its width, w, in centimetres?

    w=12\div4=3

  4. Check: 80-3\times4=80-12=68\text{ cm}^2
  5. Answer: w=3\text{ cm}

Bags, tins and packs

To find how many bags, tins or packs you need, divide the area by the area that one of them covers.

If the answer is not a whole number, round up. For an area of 50\text{ m}^2 and bags that each cover 8\text{ m}^2, 50\div8=6.25, so you need 7 bags. Six bags would leave part of the area uncovered.

Then finish the question that was asked: the number of bags, the total cost, or yes or no with the numbers that show it.

Worked example 6: Working out the cost of flooring

The diagram shows the plan of an L-shaped floor. Flooring is sold in packs. One pack covers 4\text{ m}^2 and costs £23. Work out the cost of the packs needed to cover the floor.

  1. Split the floor into two rectangles. Bottom rectangle: 12\times4=48\text{ m}^2
  2. The top rectangle is 6-4=2 m high, so its area is 7\times2=14\text{ m}^2
  3. Floor: 48+14=62\text{ m}^2, and 62\div4=15.5

  4. Your turn. How many packs must be bought?

    Buy 16 packs, because 15 packs cover only 60\text{ m}^2

  5. Cost: 16\times\pounds 23
  6. Answer: £368

Common mistakes

  • Multiplying all the lengths together, or adding them. Split the shape into rectangles and multiply only the length and width of each one.
  • Using the wrong height for a triangle. Use the height of the triangle itself, not the height of the whole shape.
  • Forgetting to halve for a triangle. The area of a triangle is half of base times height.
  • Using the diameter in \pi r^2. Halve the diameter first to get the radius.
  • Rounding down the number of bags or tins. Always round up, or part of the area is left uncovered.
  • Stopping at the area. Finish the question: the number of tins, the cost, or yes or no with your figures.

Practice questions

More on this topic: Area of Compound Shapes worksheet with full solutions