Sector Area Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sector Area.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The area of a 'pie slice' region of a circle, bounded by two radii and the arc between them.

Imagine cutting a pizza into slices. Each slice is a sector. If you cut the pizza into 4 equal slices (90° each), each slice has \frac{1}{4} of the pizza's total area. The sector area is simply the fraction of the full circle determined by the central angle, applied to the total area.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Sector area is a fraction of the circle's total area, proportional to the central angle.

Common stuck point: Like arc length, make sure the angle units match the formula. The radian form (\frac{1}{2}r^2\theta) is simpler for calculus applications.

Worked Examples

Example 1

easy
Find the area of a sector of a circle with radius 8 cm and central angle 90°.

Solution

  1. 1
    Step 1: Write the sector area formula in degrees: A = \frac{\theta}{360°} \times \pi r^2.
  2. 2
    Step 2: Substitute \theta = 90° and r = 8 cm: A = \frac{90}{360} \times \pi (8)^2.
  3. 3
    Step 3: Simplify the fraction: \frac{90}{360} = \frac{1}{4}, and r^2 = 64.
  4. 4
    Step 4: Compute: A = \frac{1}{4} \times 64\pi = 16\pi \approx 50.27 cm².

Answer

A = 16\pi \approx 50.27 cm²
A 90° sector is one-quarter of the full circle. One-quarter of the circle's area \pi(8)^2 = 64\pi cm² gives 16\pi cm². This matches \frac{1}{4}\pi r^2 for a quarter-circle.

Example 2

medium
A sector has a central angle of \frac{2\pi}{3} radians and a radius of 9 cm. Find its area.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A pizza slice (sector) has a radius of 12 inches and a central angle of 45°. Find the area of the slice.

Example 2

hard
A sprinkler rotates through an angle of 120° and waters grass up to a radius of 15 ft. What area of grass does it water? If the water only reaches between 10 ft and 15 ft from the sprinkler, what annular sector area gets watered?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

area of circlecentral angle