Area of Circles and Composite Figures Revision Notes | GCSE AQA Higher Maths
To find the area of a circle, use A = πr², squaring the radius first and then multiplying by π. Free GCSE AQA Maths notes and 11 questions answered.
Updated 25 Mar 2026 · Revise Right Team
Area of Circles and Composite Figures is part of Mensuration and calculation in GCSE AQA Higher Maths.
4 min
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42
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GCSE AQA Higher
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What this lesson covers
To find the area of a circle, use A = πr², squaring the radius first and then multiplying by π. Free GCSE AQA Maths notes and 11 questions answered.
Apply the formula $A = \pi r^2$ to calculate the area of a circle
Identify the radius from a given diameter and use it correctly in the area formula
Calculate the area of composite shapes by combining and subtracting areas of circles and rectilinear shapes
Leave answers in exact form (in terms of $\pi$) or round to a specified degree of accuracy
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Use this lesson to stay inside Mensuration and calculation while connecting the detail back to the wider Geometry and measures topic.
Area of a circle: The space enclosed by a circle, calculated using $A = \pi r^2$, where $r$ is the radius. Used whenever you need to find the surface covered by a circular region.
Radius: The distance from the centre of a circle to its edge. Always half the diameter. Used as the key measurement in the area formula.
Composite shape: A shape made by combining or subtracting two or more standard shapes (e.g. a rectangle with a semicircle removed). Area is found by adding or subtracting the individual areas.
Exact form: An answer left as a multiple of $\pi$ (e.g. $25\pi$ cm²) rather than converted to a decimal. Required when a question says 'give your answer in terms of $\pi$'.
Revision notes
Read the core explanations from Area of Circles and Composite Figures before testing yourself from memory.
1. Why Circle Area Matters
Circular regions appear everywhere in GCSE problems — from the cross-section of a pipe to a lawn surrounding a pond. Knowing how to calculate their area, and how to combine that with other shapes, unlocks a whole family of exam questions worth several marks each.
The good news is that the method is consistent every time: one formula, applied carefully, with attention to whether you have the radius or the diameter. Get that distinction right and the rest follows.
2. The Area Formula — What It Means and When to Use It
Think of the area of a circle as the number of unit squares you could tile inside it. Because a circle has no straight edges, we need a formula rather than simple counting — and that formula involves $\pi$, the ratio of a circle's circumference to its diameter ($\pi \approx 3.14159...$).
The order of operations matters here: square $r$ first, then multiply by $\pi$. Writing $\pi r^2$ does not mean $\pi r$ all squared — only $r$ is squared.
Now that the formula is pinned down, here is a fully worked example to see it in action — including how to handle both exact and decimal answers.
Worked example: Calculating the Area of a Circle (calculator-allowed exam question)
A circular table top has a diameter of 90 cm. Calculate its area. Give your answer to 1 decimal place.
$r = 90 \div 2 = 45$ cm — The question gives the diameter. Always halve it to get the radius before using the formula. [M1 — correct identification of radius]
$A = \pi r^2$ — State the formula explicitly — this earns a method mark on GCSE Higher exam papers.
$A = \pi \times 45^2$ — Substitute $r = 45$. Do not multiply by $\pi$ yet — square first. [M1 — correct substitution]
$45^2 = 45 \times 45 = 2025$ — Expand the square longhand to avoid errors. $45 \times 45 = 2025$.
$A = \pi \times 2025 = 6361.725...$ cm² — Use the $\pi$ button on your calculator for full accuracy. Do not round yet.
$A \approx 6361.7$ cm² — Round to 1 decimal place as instructed. Always include units — missing units can cost a mark. [A1 — correct answer with units]
Answer: $A \approx 6361.7$ cm²
3. Exact Answers — Leaving Results in Terms of $\pi$
Some exam questions — particularly In questions without a calculator — ask you to leave your answer in terms of $\pi$. This means you do not convert to a decimal; you write the number multiplied by $\pi$ as your final answer.
For example, if $r = 7$ cm, then $A = \pi \times 7^2 = \pi \times 49 = 49\pi$ cm². That is the exact answer. Writing $153.9...$ cm² would not be accepted when exact form is required.
Worked example: Finding Area in Exact Form (question without a calculator)
A circle has radius 9 m. Find its area, giving your answer in terms of $\pi$.
$A = \pi r^2$ — State the formula. On a question without a calculator, showing this earns a method mark.
$A = \pi \times 9^2$ — The radius is given directly — no halving needed here. [M1]
$9^2 = 9 \times 9 = 81$ — Expand longhand: $9 \times 9 = 81$. This is the step where errors creep in if you rush.
$A = 81\pi$ m² — Leave as $81\pi$ — do not press the $\pi$ button and convert. Include units. [A1]
Answer: $A = 81\pi$ m²
4. Composite Shapes — Adding and Subtracting Areas
A composite shape is built from two or more standard shapes joined together or cut from one another. The strategy is always the same: break the shape into parts you recognise, calculate each area separately, then add or subtract as the diagram requires.
Circle-based composite shapes usually involve one of two scenarios: a semicircle or quarter-circle added to a rectangle, or a circular hole removed from a larger shape. Identifying which scenario you have is the first step before any calculation.
In the diagram above, the total area is the rectangle area plus the semicircle area. A semicircle is exactly half a full circle, so its area is $\frac{1}{2}\pi r^2$. The radius of the semicircle equals half the width of the rectangle.
Worked example: Area of a Composite Shape: Rectangle Plus Semicircle
A shape consists of a rectangle measuring 8 cm by 5 cm, with a semicircle of diameter 8 cm attached to one of the longer sides. Calculate the total area. Give your answer to 1 decimal place.
Total area = Area of rectangle + Area of semicircle — Write this out before calculating. It keeps your method clear and earns a structure mark.
$r = 8 \div 2 = 4$ cm — The diameter of the semicircle equals the width of the rectangle (8 cm), so the radius is 4 cm.
$A_{\text{semi}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times \pi \times 4^2$ — A semicircle is half a full circle. Square the radius first: $4^2 = 4 \times 4 = 16$. [M1]
$A_{\text{semi}} = \frac{1}{2} \times \pi \times 16 = 8\pi \approx 25.132...$ cm² — Keep the unrounded value for now — do not round mid-calculation.
$A_{\text{total}} = 40 + 25.132... = 65.132...$ cm² — Use the full decimal from step 5, not a rounded version.
$A_{\text{total}} \approx 65.1$ cm² — Round to 1 decimal place as instructed. Include units. [A1]
Answer: $A_{\text{total}} \approx 65.1$ cm²
Now consider the subtraction scenario — a circular hole cut from a rectangle. The method mirrors what you just did, but you subtract the circle's area instead of adding it.
Worked example: Area of a Composite Shape: Rectangle with Circular Hole
A rectangular metal plate measures 12 cm by 7 cm. A circular hole of radius 2 cm is cut from its centre. Calculate the remaining area. Give your answer to 2 decimal places.
Remaining area = Area of rectangle $-$ Area of circle — State the strategy first. Subtraction problems are common in exam questions about shaded regions.
$A_{\text{circle}} = 4\pi \approx 12.566...$ cm² — Keep the full decimal — do not round yet.
$A_{\text{remaining}} = 84 - 12.566... = 71.433...$ cm² $\approx 71.43$ cm² — Subtract using the unrounded circle area, then round at the very end. [A1]
Answer: $A_{\text{remaining}} \approx 71.43$ cm²
5. Deliberate Practice
The following problems cover the full range of question types you will meet in the exam — from a straightforward circle area to a multi-step composite shape. Work through each one before checking the solution.
6. Exam Technique and Common Mistakes
With the methods secure, the final step is learning to apply them under exam conditions. The mistakes below are not careless slips — they are predictable patterns that appear in mark schemes year after year.
Every spec requirement in this lesson — applying $A = \pi r^2$, working with exact form, and calculating composite areas by addition and subtraction — has now been taught, modelled, and practised. The formula bank above gives you a compact reference to return to during revision.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-25.
Use A = πr², squaring the radius first and then multiplying by π. A table top with diameter 90 cm has radius 45 cm, so A = π × 2025 ≈ 6361.7 cm² to 1 decimal place.
What is the area of a semicircle?
A semicircle is exactly half a full circle, so its area is 1/2πr². A semicircle with diameter 8 cm has radius 4 cm, so its area is 1/2 × π × 16 = 8π ≈ 25.132 cm².
What is the area of a quarter circle?
The area of a quarter circle is ¼πr², a quarter of the full circle area. In the same way, a semicircle is ½πr², so a semicircle of radius 4 cm has area ½ × π × 16 = 8π cm².
What is a composite shape?
A composite shape is built from two or more standard shapes joined together or cut from one another. Its area is found by breaking it into parts that are known shapes, calculating each area, then adding or subtracting as the diagram requires.
How do you give the area of a circle in terms of π?
Square the radius and write the result multiplied by π, without converting to a decimal. A circle with radius 9 m has area π × 81 = 81π m², which is the exact answer.
How do you find the area of a rectangle with a semicircle on top?
Add the area of the rectangle to the area of the semicircle, 1/2πr². For an 8 cm by 5 cm rectangle with a semicircle of diameter 8 cm, the area is 40 + 8π ≈ 65.1 cm² to 1 decimal place.
How do you find the area of a shape with a circular hole?
Subtract the area of the circle from the area of the larger shape. A 12 cm by 7 cm plate with a circular hole of radius 2 cm has a remaining area of 84 - 4π ≈ 71.43 cm² to 2 decimal places.
What is the area of a circle with diameter 90 cm?
A circle with diameter 90 cm has area about 6361.7 cm². The radius is 90 ÷ 2 = 45 cm, 45² = 2025, and π × 2025 = 6361.725..., rounded to 1 decimal place.
What is the area of a circle with radius 9 m in terms of π?
The area of a circle with radius 9 m is 81π m² in terms of π. Using A = πr², 9² = 9 × 9 = 81, and the π is left in rather than converted to a decimal.
Does πr² mean (πr)²?
πr² does not mean (πr)²: only r is squared. The radius is squared first, then multiplied by π, so for r = 4, πr² = π × 16 = 16π.
When should you round in a composite area question?
In a composite area question, rounding should happen only at the very end, using unrounded values for each part. For a 12 cm by 7 cm plate with a hole of radius 2 cm, 84 - 12.566... = 71.433..., which is then rounded to 71.43 cm².