Roll a wheel once
Tonight's idea
Put a dab of paint on a wheel where it touches the ground. Roll the wheel one full turn, until the paint touches down again. The wheel has laid its whole rim along the ground, so the distance rolled equals the distance round the wheel. That distance is the circumference, \(C\).
A wheel 10 cm across rolls about 31.4 cm. One 20 cm across rolls about 62.8 cm. Divide distance by diameter: about 3.14 both times.
Why the same number? Every circle is a scaled copy of every other circle. Scaling multiplies the diameter and the circumference by the same factor, so dividing one by the other gives the same answer. That answer is called π (the Greek letter "pi"). It is 3.14159…, and its decimals never end.
\(C = \pi \times d\). The diameter is two radii, so also \(C = 2 \times \pi \times r\).
The area inside the circle also uses π. You met that in Why a pizza is πr².
Watch one: a bike wheel
A bike wheel is 70 cm across. Use \(\pi \approx 3.14\) (≈ means "is about").
One turn: \(C = 3.14 \times 70 = 219.8\) cm, about 2.2 m.
How many turns in 1 km? That is 100,000 cm, and \(100{,}000 \div 219.8 \approx 455\) turns.
Going backwards works too. If a wheel rolls 157 cm in one turn, its diameter is \(157 \div 3.14 = 50\) cm.
Roll wheels of different sizes along a ruler: mathdetours.com/demos/wheel.html.
Your turn
Use \(\pi \approx 3.14\).
- Find the circumference of a circle with (a) diameter 10 cm; (b) diameter 25 cm; (c) radius 3 m.
- (a) A tape measure wrapped round a tree trunk reads 314 cm. How thick is the trunk? (b) A round pond is 62.8 m round the edge. What is its radius?
- A scooter wheel is 20 cm across. (a) How far does it roll in one turn? (b) How far in 500 turns, in metres? (c) About how many turns does it make in 100 m?
- A big wheel is three times as wide as a small wheel. Both roll along the same 30 m path. The small wheel turns 60 times. (a) How many times does the big wheel turn? Explain without finding either diameter. (b) Now find the small wheel's diameter, to the nearest centimetre.
Stretch: one big arch or three small ones
A and B are 12 m apart. One path from A to B is a half circle with diameter 12 m. Another is three half circles in a row, each with diameter 4 m. Which path is shorter? What if there were 100 tiny half circles?
Answers
- (a) \(3.14 \times 10 = 31.4\) cm. (b) \(3.14 \times 25 = 78.5\) cm. (c) The diameter is 6 m, so \(3.14 \times 6 = 18.84\) m.
- (a) \(314 \div 3.14 = 100\) cm, so the trunk is 1 m thick. (b) Diameter \(62.8 \div 3.14 = 20\) m, so the radius is 10 m.
- (a) \(3.14 \times 20 = 62.8\) cm. (b) \(500 \times 62.8 = 31{,}400\) cm, which is 314 m. (c) 100 m is 10,000 cm, and \(10{,}000 \div 62.8 \approx 159\) turns.
- (a) 20 turns. Three times the diameter means three times the circumference, so each turn of the big wheel covers as much ground as three turns of the small one: \(60 \div 3 = 20\). (b) One turn of the small wheel covers \(30 \div 60 = 0.5\) m, which is 50 cm. Diameter \(50 \div 3.14 \approx 15.9\), so 16 cm.
Stretch. Neither. Half a circle is half of \(\pi \times d\). The big arch is \(\tfrac12 \times \pi \times 12 = 6\pi\) m. Each small arch is \(\tfrac12 \times \pi \times 4 = 2\pi\) m, and three of them make \(6\pi\) m, about 18.84 m either way. With 100 half circles the diameters still add up to 12 m, so the total is still \(6\pi\) m.