A slice of the border
Length of an arc
Half a circle
A circle of radius r has circumference 2πr. An arc is a piece of the border. Reflect the circle in a diameter: the two halves swap places, so they are equal.
A semicircle has radius r. How long is its curved border?
What this lesson covers
The idea
An arc that subtends θ° at the centre of a circle of radius r has length 2πr × θ°/360°; by symmetry a semicircle has length πr and a quarter circle πr/2.
Half a circle
A circle of radius r has circumference 2πr. An arc is a piece of the border. Reflect the circle in a diameter: the two halves swap places, so they are equal.
A semicircle has radius r. How long is its curved border?
- πr
- 2πr
- πr/2
Turn the angle
Turn the angle θ at the centre. The orange arc is also laid out straight under the circle, so you see what part of the whole circumference it is.
A share of the circumference
An arc that subtends θ° at the centre of a circle of radius r has length 2πr × θ°/360°. A semicircle has length πr and a quarter circle has length πr/2.
The two halves of a circle are equal, so a semicircle is 2πr ÷ 2 = πr. A quarter-turn moves each quarter circle onto another one, so a quarter circle is 2πr ÷ 4 = πr/2. The same idea works for any angle: the arc is θ/360 of the whole circumference.
Notes
An arc that subtends θ° at the centre of a circle of radius r has length 2πr × θ°/360°. A semicircle has length πr, a quarter circle πr/2.
Check yourself
A circle has radius 10 cm. Use π = 3.14. What is the length of a semicircle?
Answer: 31.4 cm
Semicircle = πr = 3.14 × 10 = 31.4 cm.
An arc subtends 60° at the centre of a circle of radius 6 cm. Use π = 3.14. How long is the arc?
Answer: 6.28 cm
Arc = 2πr × 60/360 = 37.68 × 1/6 = 6.28 cm.
An arc of a circle of radius 6 cm is 9.42 cm long. Use π = 3.14. What angle does it subtend at the centre? (in degrees)
Answer: 90 °
9.42 ÷ 37.68 = 1/4 of the circle, so θ = 1/4 of 360° = 90°.
The curved part of a running track is a semicircle. Two runners go round it in lanes of different radius. Who runs farther on the curve?
- The runner in the outer lane, whose semicircle has a larger radius — correct. Yes. A semicircle has length πr, so a bigger radius means a longer curve. This is why staggers are needed.
- Both run the same distance, since both run a semicircle. Both run a semicircle, but a bigger semicircle is longer: its length is πr.
- The runner in the inner lane. The inner runner has the smaller radius, so the shorter curve.