7 min readConcept sheet1-mark MCQ; 2-mark clock or pendulum question; perimeter of a sector

Length of an arc

An arc is a piece of the circumference. For a central angle θ, arc length is (θ/360) × 2πr: the same fraction of the circle as θ is of 360°.

Arc lengthl = (θ/360) × 2πr

Minor and major arcThe minor arc uses θ; the major arc uses 360° − θ

Perimeter of a sector2r + (θ/360) × 2πr

ClockThe minute hand turns 6° every minute (360° ÷ 60)

Why the formula works

A full turn at the centre is 360° and gives the whole circumference 2πr. So an angle of 1° gives 2πr/360 of arc. An angle of θ gives (θ/360) × 2πr. For example, 90° is a quarter turn, so its arc is a quarter of the circumference. The angle θ is the central angle: the angle the arc makes at the centre.

Minor arc vs major arc

Minor arcThe shorter arc between two points. Its central angle θ is less than 180°. Length = (θ/360) × 2πr.

Major arcThe longer arc between the same two points. Its central angle is 360° − θ. Length = circumference − minor arc.

Minor arc + major arc = the whole circumference.

Arc length from the angle

Problem

Find the length of an arc of a circle of radius 14 cm whose central angle is 45°. (Use π = 22/7)

Working

Formula: l = (θ/360) × 2πr.
Circumference = 2 × (22/7) × 14 = 88 cm.
θ/360 = 45/360 = 1/8.
l = (1/8) × 88 = 11 cm.

Answer

11 cm

Word problem: tip of a minute hand

Problem

The minute hand of a clock is 10.5 cm long. How far does its tip move in 20 minutes? (Use π = 22/7)

Working

In 20 minutes the hand turns 20 × 6° = 120°.
Circumference = 2 × (22/7) × 10.5 = 66 cm.
θ/360 = 120/360 = 1/3.
Distance moved = (1/3) × 66 = 22 cm.

Answer

22 cm

Finding the angle from the arc

Problem

An arc of length 44 cm is cut from a circle of radius 42 cm. Find its central angle. (Use π = 22/7)

Working

Circumference = 2 × (22/7) × 42 = 264 cm.
(θ/360) × 264 = 44.
θ = 44 × 360 ÷ 264 = 60°.

Answer

60°

In real life

The tip of a car wiper or the bob of an old wall-clock pendulum moves along an arc. Railway engineers mark the length of a curved track as an arc length.

Exam angle

1 marks: “Find the length of the arc of a circle of radius 7 cm with central angle 90°. (Use π = 22/7)” Circumference = 44 cm. Arc = (90/360) × 44 = 11 cm.

2 marks: “A pendulum 35 cm long swings through an angle of 72°. Find the length of the arc its tip describes. (Use π = 22/7)” 2πr = 2 × (22/7) × 35 = 220 cm. Arc = (72/360) × 220 = (1/5) × 220 = 44 cm.

3 marks: “Find the perimeter of a sector of a circle of radius 28 cm with central angle 45°. (Use π = 22/7)” Arc = (45/360) × 2 × (22/7) × 28 = (1/8) × 176 = 22 cm. Perimeter = 2 × 28 + 22 = 78 cm.

Remember it

Arc = the angle's share of the circumference: θ/360 of 2πr.