⭕ Circumference

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

Parts of a circle: radius, diameter, chord, arc, sector
Calculate circumference using C = πd or C = 2πr
Find radius or diameter from circumference
Calculate arc length as a fraction of the circumference
Perimeter of composite shapes involving circles

📖 Learn

1. Parts of a Circle

Radius (r): distance from centre to edge

Diameter (d): distance across through centre = 2r

Chord: straight line joining two points on the circumference

Arc: part of the circumference

Sector: "pizza slice" — bounded by two radii and an arc

Segment: region between a chord and the arc

💡 Diameter = 2 × radius. If you know one, you know the other.

2. Circumference Formulae

C = πd or C = 2πr

π ≈ 3.14159… Use π button on calculator, or π ≈ 3.142 to 3 d.p.

Example: r = 5 cm → C = 2 × π × 5 = 10π ≈ 31.4 cm

Example: d = 14 cm → C = π × 14 ≈ 44.0 cm

💡 Leave answers in terms of π for exact answers (e.g. 10π cm). Use decimals for approximate.

3. Finding Radius from Circumference

Rearrange: C = 2πr → r = C ÷ (2π)

Or: C = πd → d = C ÷ π

Example: C = 50 cm → r = 50 ÷ (2π) ≈ 7.96 cm

Example: C = 62.8 cm → d = 62.8 ÷ π ≈ 20 cm, so r ≈ 10 cm

4. Arc Length

An arc is a fraction of the full circumference, based on the sector angle θ.

Arc length = (θ / 360) × 2πr

Example: Sector with r = 6 cm, angle = 120°

Arc = (120/360) × 2π × 6 = ⅓ × 12π = 4π ≈ 12.6 cm

💡 A semicircle has arc = (180/360) × 2πr = πr

5. Perimeter of Composite Shapes

Shapes can combine straight edges with circular arcs.

Semi-circle on a rectangle: Perimeter = 3 straight sides + πr (half circumference)

Example: Rectangle 8×5 with semicircle on the 8 cm side: P = 5+5+8+π×4 = 18+4π ≈ 30.6 cm

Running track: two straight sections + two semicircles = two straights + one full circle

⚠️ When a curved edge REPLACES a straight edge, do NOT include that straight edge in the perimeter.

✏️ Worked Examples

Example 1 – Circumference from Radius

Find the circumference of a circle with radius 7 cm. Give your answer (a) in terms of π (b) to 3 s.f.

(a) C = 2πr = 2 × π × 7 = 14π cm

(b) 14π ≈ 14 × 3.14159 ≈ 44.0 cm

Example 2 – Radius from Circumference

A circle has circumference 94.2 cm. Find the radius.

r = C ÷ (2π) = 94.2 ÷ (2π) ≈ 94.2 ÷ 6.283 ≈ 15.0 cm

Example 3 – Arc Length

A sector has radius 9 cm and angle 80°. Find the arc length to 2 d.p.

Arc = (80/360) × 2π × 9 = (2/9) × 18π = 4π ≈ 12.57 cm

Example 4 – Perimeter of Composite Shape

A running track has two straight sections of 100 m each and two semicircles of radius 40 m. Find the total perimeter.

Two straights: 2 × 100 = 200 m

Two semicircles = one full circle: 2π × 40 ≈ 251.3 m

Total: 200 + 251.3 ≈ 451.3 m

🎨 Visualizer

⭕ Circle Calculator

Radius (cm):

🍕 Arc Length Calculator

Radius (cm): Angle (°):

❓ Find the Radius Quiz

cm (to 2 d.p.)

Exercise 1 – Circumference from Radius or Diameter

Give answers to 1 decimal place (use π ≈ 3.14159).

Exercise 2 – Radius or Diameter from Circumference

Round to 2 decimal places.

Exercise 3 – Arc Length

Arc = (θ/360) × 2πr. Give answers to 2 d.p.

Exercise 4 – Perimeter of Composite Shapes

Round to 1 decimal place.