Practice Sector Area in Math

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

Worksheet PDF Free Answer-key PDF Family

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°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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

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 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

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.

Example 7

easy
Find the area of a 180° sector (semicircle) in a circle of radius 4 (in terms of π\pi).

Example 8

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

Example 9

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

Example 10

easy
Find the area of a 90° sector in a circle of radius 6 (in terms of π\pi).

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 π\pi.

Example 12

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 13

easy
What fraction of a full circle is a 45°45° sector?

Example 14

easy
A 120° sector in a circle of radius 3. Find its area in terms of π\pi.

Example 15

medium
A circle of radius 10 has a sector of area 25π25\pi. What fraction of the circle is it, and what is its angle?

Example 16

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

Example 17

medium
A sector has area 6π6\pi in a circle of radius 6. Find its central angle.

Example 18

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 19

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 20

medium
Find the area of a sector with radius 1212 and central angle π3\frac{\pi}{3} radians, in terms of π\pi.
← Back to Sector Area Worked Examples Formula Explained