Circumference

Distance around a circle

The circumference is the perimeter of a circle: the distance all the way around.

Discovering pi

Measure the circumference and diameter of any round object (a can, a plate, a coin). Divide C by d. You’ll always get about 3.14. That number is pi (π).

(Small differences come from measurement error.)

The formulas

C = πd or C = 2πr

where d is the diameter and r is the radius.

Examples

Diameter given: d = 12 cm

C = π × 12 ≈ 37.7 cm

Radius given: r = 5 m

C = 2 × π × 5 = 10π ≈ 31.4 m

Working backwards

Find the diameter from C: d = C ÷ π

Example: A tree trunk has circumference 220 cm. Diameter?

d = 220 ÷ π ≈ 70 cm

Exact vs approximate answers

• 10π m is the exact answer.
• 31.4 m is an approximation.

Use the π key on a calculator for accuracy, and round at the end.

Arc length and semicircles

An arc length is a fraction of the circumference.

• A semicircle‘s curved edge is half: πd/2.
• An arc with central angle θ: (θ/360) × 2πr.

Example: Arc with angle 90° and r = 8 cm: (90/360) × 2π(8) = 4π ≈ 12.6 cm.

Perimeter of a semicircle shape (curved part + straight diameter): d = 10 → 5π + 10 ≈ 25.7.

Real life: wheels

A bike wheel with diameter 70 cm travels one circumference per turn: 70π ≈ 220 cm. In 500 turns: 500 × 2.2 m = 1,100 m.

Worked examples

Example 1: C for d = 9 in? 9π ≈ 28.3 in.

Example 2: C for r = 2.5 cm? 5π ≈ 15.7 cm.

Example 3: A circular track is 400 m around. Its diameter? 400 ÷ π ≈ 127 m.

Common mistakes

• Using the radius in C = πd (forgetting to double).
• Confusing circumference with area (πr²).
• Forgetting the straight edge when finding a semicircle’s perimeter.

Try it yourself

1. C for d = 20 cm
2. C for r = 7 m
3. Diameter if C = 50 cm
4. Perimeter of a semicircle with diameter 8 cm (include the straight edge)

Answers: 1) ≈ 62.8 cm 2) ≈ 44.0 m 3) ≈ 15.9 cm 4) ≈ 20.6 cm

Quick recap

• Circumference C = πd = 2πr.
• Pi is C ÷ d for every circle.
• Arc length is a fraction of the circumference.