MathBored

Essential Math Primer
← Back to Primer Overview
R70 • Lesson 100 of 105

Arc Length & Sector Area

Calculating arc lengths and areas of circle sectors using radians

Reserve & Extensions • K-12

Prerequisites: H44, E09

Key Concepts

  • arc length
  • sector area
  • radians
  • circle geometry

Arc Length and Sector Area

When you slice a pizza, you create a sector -- a "pie slice" shape bounded by two radii and an arc. The formulas for arc length and sector area are beautifully simple in radians, and they appear everywhere from engineering to satellite coverage calculations.

Arc Length

The arc length s is the distance along the curved edge of a sector:

s = r · θ     (θ in radians)

This formula is actually the definition of a radian: an angle of 1 radian subtends an arc exactly equal to the radius.

If the angle is given in degrees, convert first:

θ (radians) = θ (degrees) × π / 180

Worked Example 1: Arc Length

Find the arc length of a sector with radius 10 cm and central angle 2.5 radians.

s = rθ = 10 × 2.5 = 25 cm

Worked Example 2: Arc Length from Degrees

Find the arc length of a sector with radius 6 inches and central angle 120 degrees.

  1. Convert: θ = 120 × π/180 = 2π/3 radians.
  2. s = 6 × 2π/3 = 4π = 12.57 inches.

Sector Area

The sector area is the area of the "pie slice":

A = (1/2) r2 θ     (θ in radians)

Think of it as a fraction of the full circle's area: if the angle is θ out of a full rotation 2π, then the fraction is θ/(2π), and the area is (θ/(2π)) × πr2 = (1/2)r2θ.

Worked Example 3: Sector Area and Arc Length Together

A sprinkler waters a sector of a lawn with radius 15 feet and a central angle of 80 degrees. Find the arc length of the watered boundary and the total area watered.

  1. Convert: θ = 80 × π/180 = 4π/9 radians.
  2. Arc length: s = 15 × 4π/9 = 60π/9 = 20π/3 = 20.94 feet.
  3. Area: A = (1/2)(152)(4π/9) = (1/2)(225)(4π/9) = 450π/9 = 50π = 157.08 sq ft.

Common Mistake

These formulas require θ in radians. If you plug in degrees, your answer will be wildly wrong. A 90-degree angle is π/2 = 1.571 radians. Using 90 instead of 1.571 gives an answer about 57 times too large.

Quick Degree Formulas

If you prefer to stay in degrees, the formulas become: Arc length = (d/360) × 2πr and Sector area = (d/360) × πr2, where d is the angle in degrees. But the radian formulas are simpler and more commonly used.

Practice Problems

1. Find the arc length: r = 8 m, θ = π/4 radians.

Solution

s = 8(π/4) = 2π = 6.28 m.

2. Find the sector area: r = 12 cm, θ = π/3 radians.

Solution

A = (1/2)(144)(π/3) = 24π = 75.40 cm2.

3. A pizza has a 14-inch diameter. Each slice has a central angle of 45 degrees. Find the arc length (crust length) and area of one slice.

Solution

Radius = 7 inches. θ = 45 × π/180 = π/4. Arc length = 7(π/4) = 7π/4 = 5.50 inches. Area = (1/2)(49)(π/4) = 49π/8 = 19.24 sq in.

4. An arc is 10 cm long on a circle with radius 4 cm. Find the central angle in radians and degrees.

Solution

θ = s/r = 10/4 = 2.5 radians. In degrees: 2.5 × 180/π = 143.24 degrees.

5. A windshield wiper is 18 inches long and sweeps through 110 degrees. What area does it clean?

Solution

θ = 110 × π/180 = 11π/18. A = (1/2)(182)(11π/18) = (1/2)(324)(11π/18) = 99π = 311.02 sq in.

Summary

Overview