Unroll the rim

The idea

Length of the arc of a sector of angle θ (in degrees) of a circle of radius r = θ/360 × 2πr, taking the whole circle of 360° to have length 2πr.

A quarter turn on a track

A round cycle track has radius 21 m. Once round the track is 2πr = 132 m (taking π = 22/7). A cyclist rides a 90° turn of the track, from A to B.

How far does the cyclist ride?

• 90 m
• 33 m
• 132 m

Mark arcs of a given length

Turn B round, or use the dial. The bar is the whole way round the circle, unrolled like a rope. The thick blue arc and the filled part of the bar have the same length.

Same idea, along the rim

Length of an arc = θ/360 × 2πr, where θ is the angle of the sector in degrees.

The whole circle, 360°, has length 2πr (its circumference). So 1° gets 2πr / 360, and θ° gets θ times that.

For the track: 1° gets 132 / 360 = 11/30 m, so 90° gets 90 × 11/30 = 33 m. That is 90/360 = 1/4 of the way round.

Notes

Length of an arc = θ/360 × 2πr, found from the circumference 2πr of the whole circle (360°).

Check yourself

A sector has angle 60° and radius 21 cm. Find the length of its arc. (Take π = 22/7)

Answer: 22 cm

Arc = 60/360 × 2 × 22/7 × 21 = 132 / 6 = 22 cm.

The minute hand of a clock is 14 cm long. How far does its tip move in 15 minutes? (Take π = 22/7)

Answer: 22 cm

15 minutes is 90°. Arc = 90/360 × 2 × 22/7 × 14 = 88 / 4 = 22 cm.

An arc of 11 cm belongs to a circle of radius 21 cm. Find the angle of its sector. (Take π = 22/7)

Answer: 30 °

11 / 132 = 1/12, so the angle is 1/12 of 360° = 30°. Check: 30/360 × 132 = 11.

In one circle, arc P subtends 40° at the centre and arc Q subtends 80°. How do their lengths compare?

• Q is twice as long as P. — correct. Yes! The arc length θ/360 × 2πr is proportional to the angle θ, so double the angle gives double the length.
• Q is four times as long as P.. Four times is for area when the radius is doubled. Here only the angle doubles, so the arc doubles.
• They are equal.. A bigger angle cuts a longer piece of the rim, so Q is longer.