Arc Length and Sector Area
Derive arc length (s = rθ in radians; s = (θ/360)·2πr in degrees) and sector area (A = (θ/360)·πr²) formulas. Solve problems requiring exact answers and decimal approximations.
A lesson of Geometry, in Circles, about 10 minutes.
By the end of this lesson, you will be able to compute arc length and sector area using degree measure, understand both the degree and radian formulations, and solve applied problems involving circular sectors.
The Big Idea
An arc is a fraction of the full circle's circumference; a sector is the corresponding "pie slice" of area. Both computations use the same proportional reasoning: the arc/sector is to the full circle as the central angle is to 360°. This proportional approach is clean, memorable, and works for any angle.
Arc Length
Arc length is the distance along the circle (in linear units, like feet or cm).
$$\ell = \frac{\theta}{360°} \cdot 2\pi r$$
where $\theta$ is the central angle in degrees and $r$ is the radius.
Proportional reasoning: $\dfrac{\ell}{2\pi r} = \dfrac{\theta}{360°}$ (arc is the same fraction of the circumference as the angle is of 360°).
Sector Area
A sector is the region bounded by two radii and the arc.
$$A_{\text{sector}} = \frac{\theta}{360°} \cdot \pi r^2$$
Proportional reasoning: the sector's area is the same fraction of the circle's area as the angle is of 360°.
Example 1 — Arc Length
Circle with radius 12, central angle 75°. Find the arc length.
$$\ell = \frac{75}{360} \cdot 2\pi(12) = \frac{75}{360} \cdot 24\pi = \frac{5}{24} \cdot 24\pi = 5\pi \approx 15.71$$
Example 2 — Sector Area
Same circle: radius 12, central angle 75°. Find the sector area.
$$A = \frac{75}{360} \cdot \pi(12)^2 = \frac{75}{360} \cdot 144\pi = \frac{5}{24} \cdot 144\pi = 30\pi \approx 94.25$$
Example 3 — Finding the Radius from Arc Length
Arc length $= 8\pi$, central angle $= 120°$. Find the radius.
$$8\pi = \frac{120}{360} \cdot 2\pi r = \frac{1}{3} \cdot 2\pi r = \frac{2\pi r}{3}$$
$$r = \frac{8\pi \cdot 3}{2\pi} = 12$$
Example 4 — Application: Pizza Slices
A pizza has diameter 16 inches and is cut into 8 equal slices. Find the arc length and area of each slice.
Radius = 8, central angle per slice = $\frac{360°}{8} = 45°$.
Arc length: $\dfrac{45}{360} \cdot 2\pi(8) = \dfrac{1}{8} \cdot 16\pi = 2\pi \approx 6.28$ inches
Sector area: $\dfrac{45}{360} \cdot \pi(8)^2 = \dfrac{1}{8} \cdot 64\pi = 8\pi \approx 25.13$ square inches
Radian Measure (Preview)
When angles are measured in radians:
$$\ell = r\theta \qquad A_{\text{sector}} = \frac{1}{2}r^2\theta$$