Arcs and Sectors of a Circle

A sector has radius 10 cm and central angle 72°.

$\text{Arc} = \frac{72}{360} \times 2\pi \times 10 = \frac{1}{5} \times 20\pi = 4\pi \approx 12.6 \text{ cm}$

Same sector: radius 10 cm, angle 72°.

$\text{Area} = \frac{72}{360} \times \pi \times 10^2 = \frac{1}{5} \times 100\pi = 20\pi \approx 62.8 \text{ cm}^2$

A car's windscreen wiper blade is 55 cm long. It sweeps through an angle of 120°. What area does it clear?

$\text{Area} = \dfrac{120}{360} \times \pi \times 55^2 = \dfrac{1}{3} \times 3025\pi \approx 3168 \text{ cm}^2 = 0.317 \text{ m}^2$

Engineers use this calculation when designing windscreens to ensure the wiper clears enough of the driver's field of view.

A clock's minute hand is 12 cm long. In 20 minutes it sweeps through $\dfrac{20}{60} \times 360° = 120°$.

Arc swept: $\dfrac{120}{360} \times 2\pi \times 12 = \dfrac{1}{3} \times 24\pi \approx 25.1 \text{ cm}$

Area swept: $\dfrac{120}{360} \times \pi \times 12^2 = 48\pi \approx 150.8 \text{ cm}^2$

A pizza has radius 15 cm and is cut into 8 equal slices (each subtends $45°$).

Arc of crust per slice: $\dfrac{45}{360} \times 2\pi \times 15 = \dfrac{15\pi}{4} \approx 11.8 \text{ cm}$

Area of each slice: $\dfrac{45}{360} \times \pi \times 15^2 = \dfrac{225\pi}{8} \approx 88.4 \text{ cm}^2$