Sector Area and Arc Length

Finding areas and lengths of circle sectors

Sector Area and Arc Length

Sector

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

It's bounded by two radii and an arc.

Arc Length

The length of the curved part of the sector.

Formula: $L = \frac{\theta}{360°} \times 2\pi r$

Or in radians: $L = r\theta$

Sector Area

The area of the "slice."

Formula: $A = \frac{\theta}{360°} \times \pi r^2$

Or in radians: $A = \frac{1}{2}r^2\theta$

Segment

The region between a chord and the arc it cuts off.

Area of segment = Area of sector - Area of triangle

Strategy

Find the fraction of the circle: $\frac{\theta}{360°}$
Multiply by the whole circle (circumference or area)

Common Sectors

Semicircle: $\theta = 180°$ → half circle
Quarter circle: $\theta = 90°$ → one-fourth circle

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the arc length of a sector with radius 6 and central angle 60°.

💡 Show Solution

Use the arc length formula: $L = \frac{\theta}{360°} \times 2\pi r$

$L = \frac{60}{360} \times 2\pi(6)$

$L = \frac{1}{6} \times 12\pi$

$L = 2\pi$

Answer: $2\pi$ (or approximately 6.28) units

2Problem 2medium

❓ Question:

Find the area of a sector with radius 8 and central angle 135°.

💡 Show Solution

Use the sector area formula: $A = \frac{\theta}{360°} \times \pi r^2$

$A = \frac{135}{360} \times \pi(8)^2$

$A = \frac{3}{8} \times 64\pi$

$A = 24\pi$

Answer: $24\pi$ (or approximately 75.4) square units

3Problem 3hard

❓ Question:

A sector has radius 10 and arc length $5\pi$ . Find the central angle and the area of the sector.

💡 Show Solution

Step 1: Find the central angle using arc length

$L = \frac{\theta}{360°} \times 2\pi r$

$5\pi = \frac{\theta}{360} \times 2\pi(10)$

$5\pi = \frac{\theta}{360} \times 20\pi$

$5 = \frac{\theta}{360} \times 20$

$5 = \frac{20\theta}{360}$

$\theta = 90°$

Step 2: Find the sector area

$A = \frac{90}{360} \times \pi(10)^2$

$A = \frac{1}{4} \times 100\pi = 25\pi$

Answer: Central angle = $90°$ , Area = $25\pi$ square units

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics