Area of an Annulus GCSE: Ring Shapes Step by Step

What does an annulus look like?

Two concentric circles (sharing a centre) enclose a ring-shaped region. The larger circle has outer radius R and the smaller circle has inner radius r. The annulus is the shaded area between them.

Real-life examples of annuli (plural of annulus): a washer, a running track viewed from above, the cross-section of a pipe, or a ring-shaped flowerbed.

Always check: R must be greater than r, and both must be positive.

How do you calculate the area of an annulus?

Method at a glance:

1. Identify R and r. Read the outer and inner radii from the question. If you are given a diameter, halve it to find the radius.
2. Square both radii. Calculate R² and r².
3. Subtract. Find R² − r².
4. Multiply by π. Area = π(R² − r²). Use π ≈ 3.142 unless the question says to leave your answer in terms of π.
5. State the units. Area is always in square units (cm², m², etc.).

Worked example 1: basic annulus

The outer radius R = 8 cm and the inner radius r = 5 cm. Find the area of the annulus. Give your answer to 3 significant figures.

• R² = 64; r² = 25
• R² − r² = 39
• Area = π × 39 = 39π ≈ 122 cm² (3 s.f.)

Worked example 2: given diameters

A circular running track has an outer diameter of 100 m and an inner diameter of 84 m.

• Outer radius R = 50 m; inner radius r = 42 m
• R² − r² = 2 500 − 1 764 = 736
• Area = π × 736 ≈ 3.142 × 736 ≈ 2 313 m² (4 s.f.)

Always halve diameters before squaring — a common error is to use the diameter directly in πd².

Worked example 3: leaving the answer in terms of π

A pipe has an outer radius of 7 cm and a wall thickness of 2 cm. Find the cross-sectional area of the pipe wall in terms of π.

• Inner radius r = 7 − 2 = 5 cm
• Area = π(7² − 5²) = π(49 − 25) = 24π cm²

This exact-form approach is often required when the question contains surds or when further calculation follows — leaving it in terms of π avoids rounding errors.

What if the question gives the width of the ring rather than both radii?

If only R and the width w of the ring are given, use r = R − w.

Example: R = 12 cm, width of ring = 3 cm. Area = π(12² − 9²) = π(144 − 81) = 63π cm².

How do you reverse the process (finding a radius from the area)?

If the area is given, solve the equation for the unknown radius.

Example: An annulus has area 75π cm². The outer radius is 10 cm. Find the inner radius.

• π(10² − r²) = 75π
• 100 − r² = 75
• r² = 25
• r = 5 cm ✓

Always check that the resulting r is less than R and is positive.

Frequently asked questions

Does the position of the hole matter — does it have to be in the exact centre?

Yes. The standard annulus formula assumes the two circles are concentric (same centre). If the smaller circle is offset, you cannot use π(R² − r²) — you would need to subtract the area of a differently positioned circle, which is beyond GCSE. All GCSE questions use concentric circles.

How is an annulus different from a circular segment?

A circular segment is the region between a chord and an arc — a "slice" cut off a circle. An annulus is a ring formed by removing a smaller circle from the centre of a larger one. They are different shapes with different formulas; the annulus formula π(R² − r²) never involves any angles.

Can I use the formula π(R + r)(R − r) instead?

Yes — because R² − r² = (R + r)(R − r) by the difference of two squares. Both forms are correct. Some students find the factorised version quicker when the numbers are large: for R = 15, r = 9, the sum is 24 and the difference is 6, so area = π × 24 × 6 = 144π.

What units should I use for the answer?

Area is always in square units. If the radii are in centimetres, the area is in cm². If the radii are in metres, use m². Never mix units — convert everything to the same unit before calculating.

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