Area of a circle, explained with examples
The area of a circle is πr². That one formula, together with the circumference 2πr, unlocks a large family of GCSE questions on semicircles, sectors, arcs and compound shapes.
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The key words
- Radius (r): the distance from the centre to the edge.
- Diameter (d): the distance across the circle through the centre, d = 2r.
- Circumference (C): the distance all the way round, the circle's perimeter.
- π (pi): the ratio of any circle's circumference to its diameter, about 3.14159.
The formulas
Area = πr² and Circumference = πd = 2πr
A simple way to keep them apart: area is measured in square units, so its formula has a square, r². Circumference is a length, so its formula has r only once. For GCSE Maths exams in 2025, 2026 and 2027, a formulae sheet is provided with the papers, covering formulae students previously had to memorise; check the sheet for your exam year, but learn both formulas anyway, because recalling it quickly saves time.
Why the area is πr²
Cut a circle into many thin sectors, like a pizza, and arrange them alternately point-up and point-down. They form a shape close to a parallelogram. Its height is the radius r, and its base is half the circumference, πr. As the slices get thinner, the shape becomes a perfect rectangle, with area πr × r = πr². This idea of slicing a shape into thin pieces is exactly how integration works at A level.
Worked examples
Example 1: area from the radius
Find the area of a circle of radius 6 cm, to 1 decimal place.
- A = π × 6² = 36π.
- A ≈ 113.1 cm².
Example 2: area from the diameter
A circular table has a diameter of 1.4 m. Find its area.
- Halve the diameter first: r = 0.7 m.
- A = π × 0.49 ≈ 1.54 m². Using 1.4 as the radius is the most common mistake.
Example 3: exact answers
Find the area and circumference of a circle of radius 5 cm, leaving your answers in terms of π.
- Area = π × 25 = 25π cm².
- Circumference = 2π × 5 = 10π cm. “In terms of π” means do not multiply out.
Example 4: working backwards
A circle has an area of 50 cm². Find its radius.
- πr² = 50, so r² = 50/π ≈ 15.92.
- r ≈ 3.99 cm.
Example 5: a semicircle
Find the area and perimeter of a semicircle of diameter 10 cm.
- Area = ½ × π × 5² = 12.5π ≈ 39.3 cm².
- Perimeter = half the circumference plus the diameter: 5π + 10 ≈ 25.7 cm. Do not forget the straight edge.
Example 6: wheel turns
A bicycle wheel has a diameter of 65 cm. How many complete turns does it make in 1 km?
- Circumference = π × 0.65 ≈ 2.042 m per turn.
- 1000 ÷ 2.042 ≈ 489.7, so 489 complete turns. Read questions carefully: “complete turns” means round down.
Example 7: best value
A 30 cm pizza costs £12 and a 40 cm pizza costs £18. The sizes are diameters. Which is better value?
- Areas: π × 15² ≈ 706.9 cm² and π × 20² ≈ 1256.6 cm².
- Area per pound: 706.9/12 ≈ 58.9 cm² and 1256.6/18 ≈ 69.8 cm².
- The 40 cm pizza gives more pizza per pound, even though it costs more. A small increase in diameter gives a big increase in area, because the radius is squared.
Rounding and accuracy
Keep the full calculator value of π and of any intermediate answers until the very end, then round as the question asks. If a question asks for an answer to 3 significant figures, an area of 113.097… becomes 113 cm². Always include units, and remember that areas use square units such as cm² or m², while lengths such as circumference use cm or m.
Sectors and arcs (Higher)
A sector is a slice of the circle between two radii, and an arc is the curved part of its edge. If the angle at the centre is θ, the sector is θ/360 of the whole circle:
- Sector area = (θ/360) × πr²
- Arc length = (θ/360) × 2πr
Example 8: a sector
Find the area and perimeter of a sector of radius 9 cm and angle 60°.
- Area = (60/360) × π × 81 = 13.5π ≈ 42.4 cm².
- Arc = (60/360) × 2π × 9 = 3π ≈ 9.42 cm. Perimeter = 9.42 + 9 + 9 ≈ 27.4 cm.
Example 9: finding the angle
A sector of radius 10 cm has an area of 50π cm². Find its angle.
- (θ/360) × 100π = 50π, so θ/360 = 1/2.
- θ = 180°: it is a semicircle.
Compound shapes
Many questions combine circles with rectangles and triangles. Split the shape into parts, find each area, and add or subtract. A shaded ring (annulus) between two circles is the area of the large circle minus the area of the small one.
Example 10: a running track
A field is a rectangle 80 m by 40 m with a semicircle on each short end. Find its area.
- The two semicircles make one full circle of diameter 40 m, so radius 20 m.
- Area = 80 × 40 + π × 20² = 3200 + 400π ≈ 4457 m².
Common mistakes
- Using the diameter instead of the radius in πr².
- Squaring πr instead of just r: πr² means π × r × r.
- Mixing up area and circumference formulas.
- Forgetting the straight edges when finding the perimeter of a semicircle or sector.
- Rounding too early in multi-step questions.
Practice questions
Try each one before opening the answer.
Q1. Find the area of a circle of radius 3 cm in terms of π.
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Q2. Find the circumference of a circle of diameter 12 cm, to 1 d.p.
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Q3. A circle has circumference 31.4 cm. Find its radius, to 1 d.p.
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Q4. Find the area of a quarter circle of radius 8 cm, in terms of π.
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Q5. Find the area of a ring between circles of radius 7 cm and 5 cm, in terms of π.
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Frequently asked questions
Is the area of a circle on the Foundation paper?
Yes. Area and circumference of circles, including semicircles, are on both tiers. Sectors and arcs are mainly Higher, though simple quarter circles can appear on Foundation.
Which value of π should I use?
Use the π button on your calculator unless the question tells you to use 3.14 or to leave the answer in terms of π.
Key takeaways
- Area = πr²; circumference = 2πr = πd.
- Halve the diameter to get the radius.
- “In terms of π” means leave π in the answer.
- Sectors and arcs use the fraction θ/360.
- Split compound shapes into simple parts.