The area of a circle is one of the most frequently tested topics at GCSE. You need to know the formula, be confident working with both radius and diameter, and handle semicircles and quarter circles. This guide covers everything from the basic formula to reverse problems where you find the radius from a given area.
What Is the Area of a Circle?
The area of a circle is the amount of space enclosed within its circumference. It depends only on the radius — the distance from the centre to any point on the edge.
The formula uses pi, the ratio of a circle's circumference to its diameter, which is approximately 3.14159. Because the formula involves squaring the radius, doubling the radius quadruples the area — a fact examiners sometimes test.
For semicircles (half circles) and quarter circles, you calculate the full circle area first and then divide by 2 or 4 respectively.
Key Formulas
Step-by-Step Method
- Check whether you are given the radius or the diameter. If given the diameter, halve it to get the radius.
- Substitute the radius into A = pi r² and calculate.
- For semicircles or quarter circles, divide the full circle area by 2 or 4.
Worked Example 1 — Foundation Level
Question: A circle has a radius of 5 cm. Find its area. Give your answer to 1 decimal place.
Working: A = pi r² A = pi × 5² A = pi × 25 A = 78.5398...
Answer: 78.5 cm² (1 d.p.)
Worked Example 2 — Higher Level
Question: A circle has an area of 154 cm². Find its radius. Give your answer to 1 decimal place.
Working: A = pi r² 154 = pi r² r² = 154 ÷ pi r² = 49.0197... r = sqrt(49.0197...) r = 7.0014...
Answer: 7.0 cm (1 d.p.)
Worked Example 3 — Exam Style
Question: A shape is made from a rectangle measuring 10 cm by 6 cm with a semicircle added to one of the shorter sides. Find the total area of the shape. Give your answer to 1 decimal place.
Working: Rectangle area = 10 × 6 = 60 cm² The semicircle has diameter 6 cm, so radius = 3 cm. Semicircle area = ½ × pi × 3² = ½ × pi × 9 = 4.5pi = 14.137... Total area = 60 + 14.137... = 74.137...
Answer: 74.1 cm² (1 d.p.)
Common Mistakes
- Using the diameter instead of the radius. The formula uses r, not d. If you are given the diameter, divide by 2 first. Using the diameter gives an answer four times too large.
- Forgetting to square the radius. Writing pi × r instead of pi × r² gives a circumference-like answer, not an area.
- Rounding pi too early. Use the pi button on your calculator or keep the full value 3.14159... until the final step to avoid inaccuracy.
- Forgetting to halve or quarter for parts of circles. A semicircle is half the area, a quarter circle is one quarter. Always divide after computing the full circle area.
Exam Tips
- If the question says "give your answer in terms of pi", leave your answer as a multiple of pi (e.g. 25pi) and do not convert to a decimal.
- For reverse problems (finding r from area), divide the area by pi first, then take the square root.
- Always state your units as cm², m², etc. — area is measured in square units.
Practice Questions
Q1 (Foundation): A circle has a diameter of 12 cm. Find its area to 1 decimal place.
Q2 (Foundation): Find the area of a quarter circle with radius 8 cm. Give your answer to 1 decimal place.
Q3 (Higher): A semicircle has an area of 100 cm². Find its radius to 2 decimal places.
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Related Topics
Summary
- The area of a circle is A = pi r², where r is the radius.
- Always check whether you have been given the radius or the diameter.
- Semicircle area is half the full circle area; quarter circle area is one quarter.
- To find the radius from the area, rearrange to r = sqrt(A ÷ pi).
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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