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17.3 Area of a Circle

GCSE Maths Foundation · Edexcel · Chapter 17: Perimeter, Area and Volume 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: using A = πr², exact answers in terms of π, working backwards and real-life problems. 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

  • Work out the area of a circle from its radius or its diameter
  • Give an area in terms of π, or estimate it with π ≈ 3.14 without a calculator
  • Find the radius of a circle when you know its area
  • Solve real-life problems such as costs, coverage and best value

Key words

  • Area: The amount of flat space inside a shape, measured in square units such as cm² or m².
  • Radius (r): Centre to edge. The area formula always uses the radius.
  • Square (verb): Multiply a number by itself: 6² = 6 × 6 = 36.
  • Square root: The inverse of squaring: √36 = 6.
  • In terms of π: An exact answer with π left in, such as 25π cm².

The area formula

Cut a circle into thin slices and lay them top-to-tail. They make a shape very close to a rectangle. Its height is the radius r, and its length is half the circumference, πr. So the area is πr × r.

Formula: Area of a circle A = π × r² (square the radius first, then multiply by π)

Worked example: given the radius

Find the area of a circle with radius 5 cm, to 1 decimal place.
A = π × 5² = π × 25 = 78.539… cm²

Answer: 78.5 cm²

When you are given the diameter

The formula needs the radius. If you are given the diameter, halve it first.

Worked example: a round table

A round table top has diameter 1.2 m. Find its area to 2 decimal places.
r = 1.2 ÷ 2 = 0.6 m
A = π × 0.6² = π × 0.36 = 1.130 97… m²

Answer: 1.13 m²

Check: Circumference is a length (cm). Area is a space (cm²). If your area has no “squared”, you have probably used the wrong formula.

Exact answers and non-calculator questions

Worked example: in terms of π

(a) Radius 9 cm: A = π × 9² = 81π cm²
(b) Diameter 14 m: r = 7, so A = π × 7² = 49π m²

Answer: (a) 81π cm² (b) 49π m²

Worked example: using π ≈ 3.14

Estimate the area of a circle with radius 10 cm. Use π = 3.14.
10² = 100, and 3.14 × 100 = 314

Answer: 314 cm²

Working backwards: radius from area

Undo the formula one step at a time: divide by π, then square root.

Formula: r = √(A ÷ π)

Worked example: a circular rug

A circular rug has area 2 m². Find its radius to the nearest centimetre.
A ÷ π = 2 ÷ π = 0.636 61…
r = √0.636 61… = 0.797 88… m

Answer: 0.80 m (80 cm)

Real-life problems

Worked example: lawn seed

A circular lawn has radius 4.5 m. One box of grass seed covers 20 m² and costs £6.99. How much will enough seed cost?
Area = π × 4.5² = 63.617… m²
Boxes: 63.617… ÷ 20 = 3.18…, so 4 boxes are needed (3 is not enough).
Cost = 4 × £6.99

Answer: £27.96

For “best value” questions about round things (pizzas, cakes, tins), compare the area you get for each £1.

Common mistakes

  • Using the diameter in πr². Halve the diameter first.
  • Working out (π × r)² instead of π × r². Square only the radius.
  • Using the circumference formula 2πr by mistake. Area units are always squared.
  • Doubling the radius and expecting the area to double. It becomes 4 times as big.
  • Rounding down when buying materials: you need enough to cover the whole area.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out the area of a circle with radius 3 cm. Give your answer to 1 decimal place.

Answer: 28.3 cm²

Working: A = π × 3² = 9π = 28.274… = 28.3 cm².

Question 2. Work out the area of a circle with diameter 14 m. Give your answer to 1 decimal place.

Answer: 153.9 m²

Working: r = 7. A = π × 7² = 49π = 153.938… = 153.9 m².

Question 3. What is the area of a circle with radius 5 cm, in terms of π? (Choose from: 10π cm², 25π cm², 5π cm², 100π cm².)

Answer: 25π cm²

Working: A = π × 5² = 25π cm².

Related lessons

Continue in the full course

This lesson is section 17.3 (EGCMF1703) 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.

 

 

 

 

 

 

 

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