Sector Area Examples: 24 Problems with Answers

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 the central angle's fraction of $360°$ applied to the circle's full area.

Common stuck point: The procedure for sector area is the easy part; the trap is using $2\pi r$ instead of $\pi r^2$ . 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

8

cm and central angle

90°

Find the area of this sector.

Answer

A = 16\pi \approx 50.27

cm²

First step

Step 1: Write the sector area formula in degrees:

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

Full solution

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

A 90° sector is one-quarter of the full circle. One-quarter of the circle's area

\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.

Find the area of this sector.

Example 3

easy

Pizza of radius

9

is cut into

6

equal slices. Find the area of one slice in terms of

\pi

Find the area of one slice (1/6 of the pizza).

Example 4

medium

Convert the angle

\frac{5\pi}{6}

radians to degrees, then find the sector area in a circle of radius

6

Convert to degrees and find the sector area.

Example 5

medium

Two sectors share the same radius

r=8

but have angles

30°

and

150°

. Find the ratio of their areas and the difference of their areas in terms of

\pi

Example 6

hard

An annular sector lies between two concentric circles of radii

5

and

8

, swept by a central angle of

90°

. Find its area in terms of

\pi

Example 7

challenge

A circle of radius

5

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

12

inches and a central angle of

45°

. Find the area of the slice.

Find the area of this pizza 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?

Example 3

easy

Find the area of a sector with radius

r=5

and central angle

72°

. Give the exact answer in terms of

\pi

Find the sector area in terms of π.

Example 4

easy

A sector has radius

r=4

and central angle

\frac{\pi}{4}

radians. Find its area in terms of

\pi

Find the sector area in terms of π.

Example 5

easy

A sector has central angle

\pi

radians and radius

10

. What is its area?

Find the area of this sector.

Example 6

medium

A sector of radius

r=6

has area

12\pi

. Find its central angle in degrees.

The sector area is 12π. Find its central angle x in degrees.

Example 7

medium

A sector with area

8\pi

has central angle

\frac{\pi}{4}

radians. Find its radius.

The sector area is 8π. Find the radius r.

Example 8

medium

A circular garden of radius

20

ft has a triangular path cut as an isosceles slice from the center with central angle

60°

. Find the area of the remaining sector after removing this

60°

pie slice, in terms of

\pi

Example 9

medium

A sector has arc length

6\pi

and radius

9

. Find its area in terms of

\pi

Arc length is 6π. Find the sector area.

Example 10

medium

A clock face has radius

10

cm. Find the area swept by the minute hand from 12:00 to 12:15, in terms of

\pi

Area swept by the minute hand from 12:00 to 12:15.

Example 11

medium

A sector with central angle

60°

has area

\frac{25\pi}{6}

. Find the radius.

The sector area is 25π/6. Find the radius r.

Example 12

hard

A circular pie of radius

9

is cut by two radii into a

40°

sector. After the slice, what is the area of the remaining piece, in terms of

\pi

Find the area of the remaining 320° piece.

Example 13

hard

A sector with central angle

60°

in a circle of radius

6

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

r

and central angle

\theta

radians. Its area equals its arc length numerically when

r=

Example 15

hard

A sector has perimeter

P=2r+r\theta

(radians). If

r=4

and the perimeter is

14

, find the central angle

\theta

and the sector area.

Sector perimeter

Example 16

challenge

A goat is tied to a corner of a rectangular barn measuring

4\times 8

m by a rope of length

10

m. The goat grazes outside the barn. Find the grazing area in terms of

\pi

, assuming the rope can wrap around adjacent corners.

Example 17

challenge

A sector with fixed perimeter

24

(two radii + arc) — find the radius that maximizes the area, and the maximum area.

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Related Concepts

Area of a Circle Central Angle Integration by Parts

Background Knowledge

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

area of circlecentral angle