Circumference, Arc Length & Sectors

C=2π r, arc = θ360 C, sector = θ360 π r^2.

The idea

A circle of radius $r$ is described by two measurements. Its circumference, the distance once around the edge, is $C = 2\pi r$, and the area it encloses is $\pi r^{2}$. The circumference grows in proportion to the radius, while the area grows with the square of the radius, so doubling the radius doubles the circumference but quadruples the area.

Pieces of a circle are measured by the angle $\theta$ they span at the center $O$. An arc $s$ is a piece of the edge, and a sector is the wedge of the disk bounded by two radii and the arc joining them. An arc or sector spanning $\theta$ degrees is the fraction $\dfrac{\theta}{360}$ of the whole circle, since $360°$ is a full turn. That fraction of the circumference is the arc's length, and the same fraction of the area is the sector's: $\text{arc} = \frac{\theta}{360}\cdot 2\pi r, \qquad \text{sector} = \frac{\theta}{360}\cdot \pi r^{2}.$

Ways to work on it

• Walkthrough. Circumference, arc length, and sector area as multiples of π.
• Practice. One circumference, area, arc, or sector — the coefficient of π.
• Hardest. A sector area, or solve back for the radius.

Not sure where to start? Take the ten-question placement test.