Compound shape questions combine two or more simple shapes and ask you to find the total area. These questions appear on both Foundation and Higher papers and test whether you can break a complex figure into parts you already know how to handle. They are worth 3-4 marks and reward clear, well-organised working.
What Is a Compound Shape?
A compound shape (also called a composite shape) is a shape made up of two or more simpler shapes joined together. Common examples include L-shapes, T-shapes, shapes with semicircular ends, and rectangles with triangles or trapeziums attached.
The strategy is always the same: split the shape into parts whose area formulas you know, calculate each area, then add them together. Sometimes a shape is easier to handle by starting with a full rectangle and subtracting a missing piece.
Both approaches — adding simpler areas or subtracting cut-outs — are equally valid. Choose whichever makes the arithmetic easier for the particular shape you are given.
Key Formulas
Step-by-Step Method
- Study the shape and decide how to split it — either into two or more simpler shapes (addition) or as a large shape with a piece removed (subtraction).
- Label the dimensions of each part clearly — draw dashed lines on the diagram to show your split.
- Calculate the area of each part using the correct formula.
- Add the areas together (if shapes are joined) or subtract (if a piece is removed).
- Include the correct square units in your final answer.
Worked Example 1 — Foundation Level
Question: An L-shaped room has overall dimensions 10 m by 8 m. A 4 m by 3 m rectangle is missing from one corner. Find the area of the room.
Working: Full rectangle area = 10 × 8 = 80 m² Missing rectangle area = 4 × 3 = 12 m² L-shape area = 80 − 12 = 68
Answer: 68 m²
Worked Example 2 — Higher Level
Question: A shape consists of a rectangle measuring 14 cm by 6 cm with a semicircle removed from one of the longer sides. The semicircle has a diameter of 6 cm. Find the shaded area to 1 decimal place.
Working: Rectangle area = 14 × 6 = 84 cm² Semicircle radius = 6 ÷ 2 = 3 cm Semicircle area = ½ × pi × 3² = ½ × pi × 9 = 4.5pi = 14.137... Shaded area = 84 − 14.137... = 69.862...
Answer: 69.9 cm² (1 d.p.)
Worked Example 3 — Exam Style
Question: A shape is made from a trapezium on top of a rectangle. The rectangle is 12 cm wide and 5 cm tall. The trapezium has parallel sides of 12 cm (bottom, shared with the rectangle) and 8 cm (top), with a perpendicular height of 4 cm. Find the total area.
Working: Rectangle area = 12 × 5 = 60 cm² Trapezium area = ½(12 + 8) × 4 = ½ × 20 × 4 = 40 cm² Total area = 60 + 40 = 100
Answer: 100 cm²
Common Mistakes
- Missing a section. When splitting a shape, make sure every part is accounted for. Label each section and tick them off as you calculate.
- Adding when you should subtract. If a piece is cut out of a larger shape, you must subtract its area — not add it. Read the question for words like "removed" or "shaded".
- Using the wrong dimensions for each part. After splitting the shape, re-read the measurements carefully. A common error is using the overall length when a sub-shape only covers part of it.
- Forgetting to use the correct formula for each sub-shape. Each part of the compound shape may require a different area formula — rectangle, triangle, trapezium, or circle.
Exam Tips
- Sketch the split on the diagram — draw a dashed line to show how you have divided the shape. This helps you and the examiner follow your method.
- If the shape has a curved part (semicircle, quarter circle), calculate the curved area separately and keep full precision until the final rounding step.
- Show each sub-area in your working before adding or subtracting. This earns method marks even if the final arithmetic is wrong.
- Check your answer is reasonable by comparing it to a simple rectangle that encloses the whole shape — your compound area should be smaller than that rectangle.
Practice Questions
Q1 (Foundation): A T-shape is formed by a rectangle 10 cm by 3 cm on top of a rectangle 4 cm by 7 cm. Find the total area.
Q2 (Foundation): An L-shape has overall dimensions 9 m by 7 m with a 5 m by 3 m rectangle removed from one corner. Find the area.
Q3 (Higher): A shape is made from a rectangle 20 cm by 10 cm with a semicircle of diameter 10 cm added to one end. Find the total area to 1 decimal place.
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Related Topics
Summary
- Compound shapes are made from two or more simpler shapes joined together.
- Split the shape into rectangles, triangles, trapeziums, or circles and calculate each area separately.
- Add areas when shapes are joined; subtract areas when a piece is removed.
- Always label your sub-shapes and show each area calculation in your working to earn method marks.
- Choose the splitting method (addition or subtraction) that makes the arithmetic simplest for the particular shape.
- Compound shape questions are worth 3-4 marks at GCSE, so showing clear, organised working is essential.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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