Arcs, sectors and turning wheels

What you will be able to do

• find the length of an arc and the area of a sector from the radius and the angle;
• find the angle turned by the hand of a clock or by a wheel;
• change revolutions per minute into radians per second, and use $v = r\omega$.

The length of an arc

In radians, $\theta = \dfrac{s}{r}$. Turned around, this gives the arc:

This simple formula is the reason for learning radians. In degrees the same arc needs an awkward factor, $s = \dfrac{\pi r \theta}{180}$ for $\theta$ in degrees.

Example. A circular running track has radius $36$ m. A runner covers the arc that subtends $\dfrac{5\pi}{6}$ at the centre. The arc is $s = 36 \times \dfrac{5\pi}{6} = 30\pi \approx 94.2$ m.

The area of a sector

A sector is the part of a circle between two radii, like a slice of a round cake.

The whole circle has the area $\pi r^2$ and the angle $2\pi$. A sector with the angle $\theta$ holds the fraction $\dfrac{\theta}{2\pi}$ of it:

Since $s = r\theta$, the area is also $A = \dfrac{1}{2} r s$. It looks like the area of a triangle with base $s$ and height $r$.

The hands of a clock

The minute hand turns $2\pi$ rad in $60$ minutes, so it turns $\dfrac{\pi}{30}$ rad, or $6^\circ$, each minute. The hour hand turns $2\pi$ rad in $12$ hours, so it turns $\dfrac{\pi}{360}$ rad, or half a degree, each minute.

Example. The minute hand of a wall clock is $14$ cm long. How far does its tip move in $25$ minutes? The angle is $25 \times \dfrac{\pi}{30} = \dfrac{5\pi}{6}$ rad, so $s = 14 \times \dfrac{5\pi}{6} \approx 36.65$ cm.

Wheels and angular speed

A wheel that rolls without slipping moves forward one circumference, $2\pi r$, in each turn. So a distance $d$ needs $\dfrac{d}{2\pi r}$ turns.

The angular speed $\omega$ (the Greek letter omega) is the angle turned each second, in radians per second. A wheel that makes $n$ revolutions each minute turns $2\pi n$ rad in $60$ s:

While the wheel turns through $\theta$, a point on its rim moves the arc $r\theta$. Divide both by the time, and the speed of the point is

Example. A pulley of radius $0.08$ m turns at $450$ revolutions per minute. Then $\omega = \dfrac{2\pi \times 450}{60} = 15\pi \approx 47.1$ rad/s. The belt on it moves at $v = 0.08 \times 47.1 \approx 3.77$ m/s.

Two pulleys joined by a belt share the speed of the belt. So $r_1 \omega_1 = r_2 \omega_2$, and the smaller pulley turns faster.

Practise with new numbers

Try it in Python

The program follows a bicycle wheel. It changes $200$ revolutions per minute into radians per second. Then it finds the speed of the bicycle with $v = r\omega$, and the distance that it covers in $10$ minutes.

Common mistakes

• Degrees in $s = r\theta$. For $r = 5$ cm and $60^\circ$, the arc is $5 \times \dfrac{\pi}{3} \approx 5.24$ cm, not $5 \times 60$ cm.
• Losing the half in the area. $A = \frac{1}{2}r^2\theta$. Without the half, a full turn would give $2\pi r^2$, twice the area of the circle.
• Revolutions per minute as radians per second. Multiply by $2\pi$ for the angle of each turn, and divide by $60$ for each second.
• Mixing units. With $r$ in cm, the arc comes out in cm and the area in cm$^2$. Change every length into one unit first.

Remember these

• $s = r\theta$ and $A = \frac{1}{2}r^2\theta = \frac{1}{2}rs$, with $\theta$ in radians.
• The minute hand turns $\frac{\pi}{30}$ rad each minute. The hour hand turns $\frac{\pi}{360}$ rad each minute.
• A rolling wheel moves $2\pi r$ in each turn.
• For $n$ revolutions each minute, $\omega = \frac{2\pi n}{60}$ rad/s, and the rim moves at $v = r\omega$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. An arc of $25$ cm lies on a circle of radius $10$ cm. What angle does it subtend at the centre?

2. Through what angle, in radians, does the minute hand turn in $20$ minutes?

3. Find the area of a sector of radius $6$ cm and angle $\dfrac{\pi}{3}$.

4. A wheel of radius $0.4$ m rolls through $10$ full turns. How far does it move?

5. Change $120$ revolutions per minute into radians per second.