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°90° each), each slice has 14\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 the central angle's fraction of 360°360° applied to the circle's full area.

Common stuck point: The procedure for sector area is the easy part; the trap is using 2πr2\pi r instead of πr2\pi r^2. Asking "Am I asked for the area of a slice of the circle (square units), not the curved edge or the whole circle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asked for the area of a slice of the circle (square units), not the curved edge or the whole circle?

Worked Examples

Example 1

easy
Find the area of a sector of a circle with radius 88 cm and central angle 90°90°.

Answer

A=16π≈50.27A = 16\pi \approx 50.27 cm²

First step

1
Step 1: Write the sector area formula in degrees: A=θ360°×πr2A = \frac{\theta}{360°} \times \pi r^2.

Full solution

  1. 2
    Step 2: Substitute θ=90°\theta = 90° and r=8r = 8 cm: A=90360×π(8)2A = \frac{90}{360} \times \pi (8)^2.
  2. 3
    Step 3: Simplify the fraction: 90360=14\frac{90}{360} = \frac{1}{4}, and r2=64r^2 = 64.
  3. 4
    Step 4: Compute: A=14×64π=16π≈50.27A = \frac{1}{4} \times 64\pi = 16\pi \approx 50.27 cm².
A 90° sector is one-quarter of the full circle. One-quarter of the circle's area π(8)2=64π\pi(8)^2 = 64\pi cm² gives 16π16\pi cm². This matches 14πr2\frac{1}{4}\pi r^2 for a quarter-circle.

Example 2

medium
A sector has a central angle of 2π3\frac{2\pi}{3} radians and a radius of 99 cm. Find its area.

Example 3

easy
Pizza of radius 99 is cut into 66 equal slices. Find the area of one slice in terms of π\pi.

Example 4

medium
Convert the angle 5π6\frac{5\pi}{6} radians to degrees, then find the sector area in a circle of radius 66.

Example 5

medium
Two sectors share the same radius r=8r=8 but have angles 30°30° and 150°150°. Find the ratio of their areas and the difference of their areas in terms of π\pi.

Example 6

hard
An annular sector lies between two concentric circles of radii 55 and 88, swept by a central angle of 90°90°. Find its area in terms of π\pi.

Example 7

challenge
A circle of radius 55 contains a regular hexagon inscribed in it. Find the total area of the six circular segments between the hexagon's sides and the circle, in exact form.

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 1212 inches and a central angle of 45°45°. Find the area of the slice.

Example 2

hard
A sprinkler rotates through an angle of 120°120° and waters grass up to a radius of 1515 ft. What area of grass does it water? If the water only reaches between 1010 ft and 1515 ft from the sprinkler, what annular sector area gets watered?

Example 3

easy
Find the area of a sector with radius r=5r=5 and central angle 72°72°. Give the exact answer in terms of π\pi.

Example 4

easy
A sector has radius r=4r=4 and central angle π4\frac{\pi}{4} radians. Find its area in terms of π\pi.

Example 5

easy
A sector has central angle π\pi radians and radius 1010. What is its area?

Example 6

medium
A sector of radius r=6r=6 has area 12π12\pi. Find its central angle in degrees.

Example 7

medium
A sector with area 8π8\pi has central angle π4\frac{\pi}{4} radians. Find its radius.

Example 8

medium
A circular garden of radius 2020 ft has a triangular path cut as an isosceles slice from the center with central angle 60°60°. Find the area of the remaining sector after removing this 60°60° pie slice, in terms of π\pi.

Example 9

medium
A sector has arc length 6π6\pi and radius 99. Find its area in terms of π\pi.

Example 10

medium
A clock face has radius 1010 cm. Find the area swept by the minute hand from 12:00 to 12:15, in terms of π\pi.

Example 11

medium
A sector with central angle 60°60° has area 25π6\frac{25\pi}{6}. Find the radius.

Example 12

hard
A circular pie of radius 99 is cut by two radii into a 40°40° sector. After the slice, what is the area of the remaining piece, in terms of π\pi?

Example 13

hard
A sector with central angle 60°60° in a circle of radius 66 has its inscribed isosceles triangle (formed by the two radii and the chord) removed. Find the area of the remaining circular segment in exact form.

Example 14

hard
A sector has radius rr and central angle θ\theta radians. Its area equals its arc length numerically when r=r= ?

Example 15

hard
A sector has perimeter P=2r+rθP=2r+r\theta (radians). If r=4r=4 and the perimeter is 1414, find the central angle θ\theta and the sector area.

Example 16

challenge
A goat is tied to a corner of a rectangular barn measuring 4×84\times 8 m by a rope of length 1010 m. The goat grazes outside the barn. Find the grazing area in terms of π\pi, assuming the rope can wrap around adjacent corners.

Example 17

challenge
A sector with fixed perimeter 2424 (two radii + arc) — find the radius that maximizes the area, and the maximum area.
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Background Knowledge

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

area of circlecentral angle