The area of a circle is π times the radius squared, A = πr², so a circle with a radius of 5 cm has an area of 25π, about 78.5 cm².
The circumference, the distance around the edge, is 2πr, or π times the diameter. In England, pupils name the radius, the diameter and the circumference in Year 6, calculate areas of circles at key stage 3, and must know both formulas by heart for GCSE, where they are not printed on the paper. This guide shows how to use them from a radius or a diameter, how to work backwards from an answer, and the slips that cost the most marks.
The formula for the area and the circumference
The AQA GCSE Mathematics specification lists the two formulas in its appendix of formulae that students are expected to know and that will not be given in the exam: where r is the radius and d the diameter, the circumference is 2πr = πd and the area is πr². Both sit in the basic Foundation content, so every GCSE candidate needs them.
The order of operations matters in πr². The index comes before the multiplication, so you square the radius first and multiply by π afterwards: for r = 3, that is 3² = 9, then 9 × π = 28.27. Squaring π × 3 instead gives 88.83, more than three times too much. If that rule feels shaky, our guide to BIDMAS and the order of operations explains it step by step.
| Given | Radius | Area, exact | Area, 2 dp | Circumference |
|---|---|---|---|---|
| radius 1 cm | 1 cm | π cm² | 3.14 cm² | 2π = 6.28 cm |
| radius 3 cm | 3 cm | 9π cm² | 28.27 cm² | 6π = 18.85 cm |
| radius 5 cm | 5 cm | 25π cm² | 78.54 cm² | 10π = 31.42 cm |
| diameter 12 m | 6 m | 36π m² | 113.10 m² | 12π = 37.70 m |
| diameter 20 cm | 10 cm | 100π cm² | 314.16 cm² | 20π = 62.83 cm |
Notice the units: the circumference is a length, in cm or m, while the area is in square units, cm² or m². Mixing them up is one of the easiest marks to lose.
Radius and diameter: which one you are given
The national curriculum programmes of study for mathematics ask Year 6 pupils to name the parts of a circle, including the radius, the diameter and the circumference, and to know that the diameter is twice the radius. At key stage 3 the same document adds perimeters of 2-D shapes including circles, areas of circles and composite shapes.
That one fact, d = 2r, is behind the most common slip. A question gives a diameter of 10 cm, and the 10 goes straight into πr²: the answer comes out as 100π, about 314 cm², when the correct radius is 5 cm and the area is 25π, about 78.5 cm². The wrong answer is exactly four times too big, because doubling the radius multiplies the area by 2 × 2 = 4.
The same rule explains a classic pizza puzzle. A pizza 12 inches across has a radius of 6 inches and an area of 36π, about 113.1 square inches. Two pizzas 8 inches across have 2 × 16π = 32π, about 100.5 square inches. The single large pizza is bigger than the two small ones together, even though 12 is less than 8 + 8.
The circumference of a circle, and working backwards
For the distance around the edge, πd is usually the quickest form. A bicycle wheel 70 cm across has a circumference of 70π, about 219.9 cm, so one full turn moves the bike about 2.2 m.
Questions often run the other way. If you know the circumference, divide by π to get the diameter: a tree trunk that measures 1.57 m round has a diameter of 1.57 ÷ π, about 0.5 m. If you know the area, undo each step of A = πr² in reverse order: divide by π, then take the square root. For an area of 50 cm², 50 ÷ π is about 15.92, and the square root of 15.92 is about 3.99, so the radius is 3.99 cm. A quick check closes the loop: π × 3.99² is about 50.0.
Working backwards is where rounding causes trouble. Keep the full calculator value until the last line, and round only the final answer to the accuracy the question asks for.
Which value of π to use
π is about 3.14159 and its decimals never end, so every value you type is an approximation. The table compares the usual choices for a circle with a radius of 10 cm, whose true area is 314.16 cm² to two decimal places.
| Value used | Area for r = 10 cm | Difference from the true area |
|---|---|---|
| 3 | 300 cm² | about 4.5% too small |
| 3.14 | 314 cm² | about 0.05% too small |
| 22/7 | 314.29 cm² | about 0.04% too big |
| π key on a calculator | 314.16 cm² | none to 2 dp |
On a calculator paper, use the π key unless the question tells you to use 3.14. Without a calculator, the AQA specification notes that solutions in terms of π may be asked for, and calculating exactly with multiples of π is part of the additional Foundation content: an answer of 25π cm² is complete and exact, with nothing left to round. Our guide to the non-calculator GCSE maths paper covers the other exact forms that paper expects.
The value 3 is still useful for one job: estimating. Three times the radius squared gives a quick lower bound, so for r = 10 cm anything below 300 cm² must be wrong.
Finding a circle's area from the diameter, step by step
When a question gives the diameter, five steps avoid nearly every slip. Halve the diameter to get the radius. Square the radius. Multiply by π. Write the unit squared. Round only at the end.
Take a plate 8 cm across. The radius is 4 cm, 4² = 16, and 16π is 50.27 cm² to two decimal places. A sense check helps: the circle fits inside a square 8 cm by 8 cm, whose area is 64 cm², and a circle always fills about 78.5% of the square around it, because π ÷ 4 is about 0.785. Here 64 × 0.785 gives 50.24, close to the exact answer, so the result is plausible.
The same check catches the diameter slip at once. If you had used 8 as the radius, you would get 201 cm², more than three times the area of the whole square, which is impossible.
Semicircles, sectors and composite shapes
The AQA specification puts areas of circles and composite shapes in the basic Foundation content, and arc lengths and areas of sectors in the additional Foundation content. Both start from the full circle and take a fraction of it.
A semicircle with a radius of 4 cm has half the area of the circle: 16π ÷ 2 = 8π, about 25.13 cm². Its perimeter is the curved half, 4π, plus the straight diameter, 8 cm, so about 20.57 cm in all; leaving out the diameter is a frequent mistake. A sector with an angle of 90° is a quarter of the circle: with a radius of 6 cm, its area is 36π ÷ 4 = 9π, about 28.27 cm². In general, a sector of angle θ has an area of θ/360 × πr².
Composite shapes combine these pieces. A running-track shape made of a 10 cm by 4 cm rectangle with a semicircle on each short end has an area of 40 cm² for the rectangle plus one full circle of radius 2 cm, 4π, about 12.57 cm²: 52.57 cm² in total. Areas of circles first appear at key stage 3, which starts in Year 7; our article on maths in Year 7 shows how that year is organised.
Frequently asked questions
What is the formula for a circle's area?
A = πr², where r is the radius. Square the radius first, then multiply by π.
How do you find the circumference from the diameter?
Multiply the diameter by π. A circle 10 cm across has a circumference of 10π, about 31.42 cm.
Is the radius half the diameter?
Yes. The diameter is twice the radius, so halve a diameter before putting it into πr².
Do you get the circle formulas in the GCSE maths exam?
No. AQA lists the circumference and area formulas among those students are expected to know, and they are not given in the exam.
Which value of pi should you use?
The π key on a calculator, unless the question says 3.14. Without a calculator, you may be asked to leave the answer in terms of π, such as 25π cm².
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