Arc Length and Sector Area

Prerequisites

Before studying this topic, make sure you understand:

• Converting between degrees and radians
• Basic circle geometry (radius, circumference, area)

Arc Length

Arc length is the distance along the curved edge of a circle between two points.

The Formula (Radians)

For a circle with radius $r$ and central angle $\theta$ (in radians):

$s = r\theta$

Where:

• $s$ = arc length
• $r$ = radius
• $\theta$ = central angle in radians

Why This Formula Works

The circumference of a full circle is $2\pi r$. A full rotation is $2\pi$ radians.

So the arc length for angle $\theta$ is: $s = \frac{\theta}{2\pi} \times 2\pi r = r\theta$

Examples

Example 1: A circle has radius 5 cm. Find the arc length for a central angle of $\frac{\pi}{3}$ radians.

$s = r\theta = 5 \times \frac{\pi}{3} = \frac{5\pi}{3} \text{ cm} \approx 5.24 \text{ cm}$

Example 2: A circle has radius 10 inches. Find the arc length for a central angle of $60°$.

First, convert to radians: $60° = \frac{\pi}{3}$ radians

$s = 10 \times \frac{\pi}{3} = \frac{10\pi}{3} \text{ inches} \approx 10.47 \text{ inches}$

Example 3: If an arc has length 12 cm and the radius is 8 cm, find the central angle in radians.

$\theta = \frac{s}{r} = \frac{12}{8} = \frac{3}{2} \text{ radians} = 1.5 \text{ radians}$

Sector Area

A sector is a "slice" of a circle, like a piece of pie.

The Formula (Radians)

For a circle with radius $r$ and central angle $\theta$ (in radians):

$A = \frac{1}{2}r^2\theta$

Where:

• $A$ = sector area
• $r$ = radius
• $\theta$ = central angle in radians

Why This Formula Works

The area of a full circle is $\pi r^2$. A full rotation is $2\pi$ radians.

So the sector area for angle $\theta$ is: $A = \frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2}r^2\theta$

Examples

Example 1: Find the area of a sector with radius 6 cm and central angle $\frac{\pi}{4}$ radians.

$A = \frac{1}{2}r^2\theta = \frac{1}{2}(6)^2 \times \frac{\pi}{4} = \frac{1}{2}(36) \times \frac{\pi}{4} = \frac{36\pi}{8} = \frac{9\pi}{2} \text{ cm}^2$

Example 2: A pizza with radius 12 inches is cut into 8 equal slices. What is the area of one slice?

Each slice has central angle: $\frac{2\pi}{8} = \frac{\pi}{4}$ radians

$A = \frac{1}{2}(12)^2 \times \frac{\pi}{4} = \frac{1}{2}(144) \times \frac{\pi}{4} = \frac{144\pi}{8} = 18\pi \text{ in}^2 \approx 56.55 \text{ in}^2$

Example 3: A sector has area $20\pi$ cm² and radius 10 cm. Find the central angle.

$\theta = \frac{2A}{r^2} = \frac{2(20\pi)}{(10)^2} = \frac{40\pi}{100} = \frac{2\pi}{5} \text{ radians}$

Combined Problems

Example: A circle has radius 15 m. A sector has central angle $\frac{2\pi}{3}$ radians. Find both the arc length and sector area.

Arc length: $s = r\theta = 15 \times \frac{2\pi}{3} = 10\pi$ m $\approx 31.42$ m

Sector area: $A = \frac{1}{2}r^2\theta = \frac{1}{2}(15)^2 \times \frac{2\pi}{3} = \frac{1}{2}(225) \times \frac{2\pi}{3} = \frac{450\pi}{6} = 75\pi$ m² $\approx 235.62$ m²

Important Notes

⚠️ These formulas only work when the angle is in radians!

If you're given degrees, convert to radians first.

Real-World Applications

• Architecture: Circular windows, arches, domes
• Engineering: Gears, pulleys, rotating machinery
• Sports: Basketball court three-point lines, running tracks
• Landscaping: Curved garden beds, irrigation coverage
• Navigation: Distance along Earth's surface (great circle routes)