19.3 Arc length and sector area — Learn

Prerequisites: use circle circumference and area formulae; calculate fractions of amounts.

Objectives: calculate arc lengths, sector areas and sector perimeters from an angle and radius.

Fractions of a circle

A sector is a slice of a circle bounded by two radii and an arc. Its angle is a fraction of $360^\circ$. The arc length and area use that same fraction.

Example 1. Find the arc length for radius 12 cm and angle $60^\circ$.

Find the fraction of the full turn: $60\div360=\frac16$.

Find one sixth of the circumference: $\frac16\times2\pi\times12=4\pi$.

The arc length is $4\pi\text{ cm}$.

Example 2. Find the area of a $90^\circ$ sector with radius 8 cm.

The sector is one quarter of a circle: $90\div360=\frac14$.

Take one quarter of the circle area: $\frac14\times\pi\times8^2=16\pi$.

So its area is $16\pi\text{ cm}^2$. Add both radii to the arc when finding sector perimeter.

Example 3. A sector has area $20\pi\text{ cm}^2$ in a circle of radius 10 cm. Work out its angle.

Find the fraction of the full circle: $\frac{20\pi}{\pi\times10^2}=\frac15$.

Find one fifth of $360^\circ$: $\frac15\times360^\circ=72^\circ$.

So the sector angle is $72^\circ$.

Watch out. The angle fraction must use degrees out of $360^\circ$. Do not use the angle alone as a multiplier.

Progress check

A quarter of the pond boundary needs edging. Work out the arc length in terms of $\pi$, then work out the perimeter of that quarter sector.

You should now be able to

• identify the fraction $\theta/360$ of a full circle;
• calculate an arc length and sector area;
• include both radii when finding sector perimeter;
• keep exact answers in terms of $\pi$.