Circles, arcs and sectors
Circles
- For a circle with radius $r$ (diameter $d = 2r$):
$$\text{circumference} = 2\pi r = \pi d, \qquad \text{area} = \pi r^2$$
- Worked example: radius $7\ \text{cm}$ → circumference $14\pi\ \text{cm}$, area $49\pi\ \text{cm}^2$.
Practice
A circle has radius 7 cm. Its area is kπ cm². What is k?
Area = πr² = π(7²) = 49π, so k = 49.
Practice
A circle has radius 7 cm. Its circumference is kπ cm. What is k?
Circumference = 2πr = 2π(7) = 14π, so k = 14.
Arcs and sectors
- A sector is a slice between two radii with angle $\theta$. It is the fraction $\dfrac{\theta}{360}$ of the circle:
$$\text{arc} = \frac{\theta}{360}\times 2\pi r, \qquad \text{sector area} = \frac{\theta}{360}\times \pi r^2$$
- Worked example: $\theta = 90^{\circ}$, $r = 8$ → fraction $\tfrac14$, arc $4\pi\ \text{cm}$, area $16\pi\ \text{cm}^2$.
Practice
A sector has angle 90° and radius 8 cm. Its area is kπ cm². What is k?
Fraction 90/360 = ¼; area = ¼ × π × 8² = 16π, so k = 16.
You've got it
Key idea
- circumference $= 2\pi r$; area $= \pi r^2$
- a sector of angle $\theta$ is the fraction $\dfrac{\theta}{360}$ of the circle
- $90^{\circ}$ sector of radius $8$: arc $4\pi$, area $16\pi$