What Is Pi? Circles and Circumference Explained — KS3 Maths

Pi (π) is the ratio of a circle's circumference to its diameter — approximately 3.14159. It is the same for every circle, no matter the size. Use the formula C = πd (or C = 2πr) to find the circumference, and A = πr² to find the area. Both formulas appear on every KS3 and GCSE maths paper.

What exactly is pi?

Measure the distance around any circle (its circumference) and divide by the distance straight across (its diameter). The answer is always the same number: π ≈ 3.14159…

This ratio never changes because all circles are the same shape — just at different scales. Pi is an irrational number, meaning its decimal goes on forever without repeating. For KS3 and GCSE, you either use the π button on your calculator or the approximation 3.14.

Key circle vocabulary

Term | Definition
Radius (r) | Distance from the centre to the edge
Diameter (d) | Distance straight across: d = 2r
Circumference (C) | The perimeter — distance all the way round
Area (A) | The space inside the circle

What is the circumference formula?

There are two equivalent forms:

C = πd (use when you know the diameter)

C = 2πr (use when you know the radius)

They give the same answer because d = 2r.

Worked example 1 — circumference from diameter

A circle has a diameter of 10 cm. Find its circumference. Give your answer to 1 decimal place.

C = πd = π × 10 = 31.415… ≈ 31.4 cm

Answer: 31.4 cm

Worked example 2 — circumference from radius

A circular pond has a radius of 4 m. Find its circumference. Give your answer to 2 decimal places.

C = 2πr = 2 × π × 4 = 25.132… ≈ 25.13 m

Answer: 25.13 m

Worked example 3 — finding the radius from the circumference

A circle has a circumference of 50 cm. Find its radius to 1 decimal place.

Rearrange C = 2πr: r = C ÷ (2π) = 50 ÷ (2 × π) = 50 ÷ 6.2831… = 7.957… ≈ 8.0 cm

Answer: r ≈ 8.0 cm

What is the area of a circle formula?

A = πr²

Always use the radius (not the diameter) for area. If you are given the diameter, halve it first.

Worked example 4 — area from radius

Find the area of a circle with radius 6 cm. Give your answer to 1 decimal place.

A = πr² = π × 6² = π × 36 = 113.097… ≈ 113.1 cm²

Answer: 113.1 cm²

Worked example 5 — area from diameter

A circular table has a diameter of 90 cm. Find its area to the nearest cm².

First, find the radius: r = 90 ÷ 2 = 45 cm

A = πr² = π × 45² = π × 2025 = 6361.73… ≈ 6362 cm²

Answer: 6362 cm²

How do you leave answers in terms of pi?

Some KS3 and GCSE questions ask for an exact answer. Instead of pressing the π button, leave π as a symbol.

Example: radius = 5 cm

Circumference: C = 2π × 5 = 10π cm (exact)

Area: A = π × 5² = 25π cm² (exact)

These are cleaner than rounding and show full mathematical precision.

Common mistakes at KS3

Mistake | Example of error | Correct approach
Using diameter instead of radius in A = πr² | A = π × 10² when d = 10 | Find r first: r = 5, then A = π × 5²
Squaring π instead of r | A = π² × r | Only r is squared: A = π × r²
Rounding too early | Using π ≈ 3.14 mid-calculation | Keep full π in your calculator until the final step
Confusing circumference and area | Giving cm instead of cm² | Circumference → cm; area → cm²

Why does pi appear in both formulas?

Both formulas come from the same definition. Because circumference = π × diameter, and a circle's area can be derived by "unrolling" it into a triangle, the ratio π shows up in both. At A-level you prove this more rigorously using calculus, but at KS3 the key point is that π links every circle measurement — once you know one dimension, π unlocks all the others.

How pi is introduced in the national curriculum

The DfE's KS3 mathematics programme of study explicitly requires students to calculate and solve problems involving the circumferences and areas of circles, introduced in Year 8 or 9 depending on the school. BBC Bitesize's KS3 maths resources confirm that π is one of the most important constants in secondary mathematics and appears across geometry, trigonometry and statistics topics right through to A-level.

Frequently asked questions

What is pi in maths?

Pi (π) is the ratio of a circle's circumference to its diameter. It is approximately 3.14159 and is the same for every circle in existence. Because it cannot be written as an exact fraction, pi is called an irrational number.

What is the circumference of a circle?

The circumference is the distance all the way around the circle — its perimeter. Use the formula C = πd (where d is the diameter) or equivalently C = 2πr (where r is the radius).

What is the formula for the area of a circle?

The area of a circle is A = πr², where r is the radius. If you are given the diameter instead, halve it to find the radius before substituting into the formula.

What is the difference between radius and diameter?

The radius is the distance from the centre of the circle to any point on its edge. The diameter is the distance straight across the circle through the centre. The diameter is always exactly twice the radius: d = 2r.

How do I find the circumference if I only know the area?

First find the radius from the area: r = √(A ÷ π). Then use C = 2πr. For example, if A = 50 cm², then r = √(50 ÷ π) ≈ 3.99 cm, and C = 2π × 3.99 ≈ 25.07 cm.

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