8.3 Area of Compound Shapes

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: finding the area of shapes made from rectangles, triangles and trapezia. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Split a compound shape into simpler shapes and add their areas
• Find an area by subtracting one shape from another
• Work out missing lengths on a compound shape
• Solve real-life problems involving area, such as paint, turf and cost

Key words

• Compound shape: A shape made by joining simpler shapes, such as rectangles and triangles.
• Split: Draw a line to cut a compound shape into simple shapes you can find the area of.
• Shaded region: The part of a diagram you are asked to find. Often big shape − small shape.
• Coverage: How much area one tin of paint, one box of seed, etc. will cover.

Splitting into rectangles

Draw a line to split the shape into rectangles. Work out any missing lengths, find each area, then add.

Worked example: an L-shape

Rectangle A: 10 × 4 = 40 cm². Rectangle B is 6 cm wide and 7 − 4 = 3 cm tall: 6 × 3 = 18 cm².
Total = 40 + 18 = 58

Answer: 58 cm²

Key fact: You can split the shape in different ways – the total area is always the same. Check: 10 × 7 − 4 × 3 = 70 − 12 = 58 cm².

Subtracting areas

When a shape has a piece cut out, find the area of the whole shape and subtract the piece.

Worked example: a lawn with a pond

Whole garden: 12 × 8 = 96 m². Pond: 5 × 3 = 15 m².
Lawn = 96 − 15 = 81

Answer: 81 m²

Shapes with triangles and trapezia

Use the right formula for each piece: rectangle l × w, triangle 1/2 × b × h, trapezium 1/2 × (a + b) × h.

Worked example: the end wall of a house

Rectangle: 8 × 5 = 40 m². Triangle: base 8 m, height 3 m: 1/2 × 8 × 3 = 12 m².
Total = 40 + 12 = 52

Answer: 52 m²

Area problems in real life

Worked example: painting the wall

The wall above (52 m²) has a window 2 m by 1 m. One tin of paint covers 12 m² and costs £15. How much does it cost to paint the wall once?
Area to paint = 52 − 2 × 1 = 50 m². Tins = 50 ÷ 12 = 4.16…, so buy 5 tins (round up).
Cost = 5 × £15 = £75

Answer: £75

Exam tip: When buying tins, boxes or packs, always round up – 4 tins would not be enough.

Common mistakes

• Using a length from the outside of the shape for a piece it does not belong to. Label each piece with its own lengths.
• Counting an overlap twice, or leaving a gap, when splitting the shape.
• Forgetting to halve for triangles inside the compound shape.
• Rounding down the number of tins or packs needed.

Revision questions

Try these, then check your answers below each one.

Question 1. An L-shape is made by cutting a 3 cm by 2 cm rectangle from one corner of an 8 cm by 6 cm rectangle. Work out its area.

Answer: 42 cm²

Working: 8 × 6 - 3 × 2 = 48 - 6 = 42 cm².

Question 2. The end of a shed is a rectangle 10 m wide and 4 m high with a triangle on top. The triangle has base 10 m and height 3 m. Work out the total area.

Answer: 55 m²

Working: 10 × 4 + 1/2 × 10 × 3 = 40 + 15 = 55 m².

Question 3. A 3 cm by 3 cm square hole is cut from a 12 cm by 9 cm rectangle. What area is left? (Choose from: 99 cm², 108 cm², 105 cm², 81 cm².)

Answer: 99 cm²

Working: 108 - 9 = 99 cm².

Related lessons

• Previous lesson: 8.2 Trapezia and Changing Units
• Next lesson: 8.4 Surface Area of 3D Solids
• In this chapter: 8.1 Rectangles, Parallelograms and Triangles
• In this chapter: 8.5 Volume of Prisms
• In this chapter: 8.6 More Volume and Surface Area

Continue in the full course

This lesson is section 8.3 (EGCMF0803) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.