Practice Sector Area in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

challenge

A sector with fixed perimeter

24

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

Example 3

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 4

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 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

challenge

A cone is made by rolling a 216° sector of radius 10 into a cone (the sector radius becomes the slant height). Find the base radius of the cone.

This sector is rolled into a cone. Find the base radius of the cone.

Example 7

easy

Find the area of a 180° sector (semicircle) in a circle of radius 4 (in terms of

\pi

Find the area of this semicircle sector.

Example 8

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 9

easy

A sector has central angle

\pi

radians and radius

10

. What is its area?

Find the area of this sector.

Example 10

easy

Find the area of a 90° sector in a circle of radius 6 (in terms of

\pi

Find the sector area in terms of π.

Example 11

medium

Find the area of the segment cut off by a 90° sector in a circle of radius 4 (sector minus triangle), in terms of

\pi

Example 12

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 13

easy

What fraction of a full circle is a

45°

sector?

Example 14

easy

A 120° sector in a circle of radius 3. Find its area in terms of

\pi

Find the sector area in terms of π.

Example 15

medium

A circle of radius 10 has a sector of area

25\pi

. What fraction of the circle is it, and what is its angle?

The sector area is 25π. What fraction of the circle is it, and what is x?

Example 16

hard

A sector has radius

r

and central angle

\theta

radians. Its area equals its arc length numerically when

r=

Example 17

medium

A sector has area

6\pi

in a circle of radius 6. Find its central angle.

The shaded sector has area 6π. Find its central angle x.

Example 18

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 19

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 20

medium

Find the area of a sector with radius

12

and central angle

\frac{\pi}{3}

radians, in terms of

\pi

Find the sector area in terms of π.

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Related Concepts

Area of a Circle Central Angle Integration by Parts