Area of a Sector: Formula and Examples

The area of a sector is the area of a pie-shaped slice of a circle, bounded by two radii and an arc. Find the fraction of the circle the sector covers, then multiply by the circle’s area. With central angle $\theta$ in degrees, the area is $\frac{\theta}{360^\circ} \cdot \pi r^2$ .

Before this lesson: You should know Circumference and Area of Circles, Arcs and Central Angles and Arc Length.

What Is a Sector?

A sector is the region inside a circle between two radii and the arc that joins them. A slice of pizza is a sector. The central angle tells you how big the slice is.

Circle with center O and radius 6. A sector with a central angle of 100 degrees is shaded orange. The sector is 100 out of 360 degrees of the circle, so its area is 100 over 360 times the circle area, 36 pi, which is 10 pi, about 31.42 square units. — The sector is 100/360 of the circle, so its area is 100/360 of the circle’s area.

The Sector Area Formula

A whole circle has 360° and area $\pi r^2$ . A sector with central angle $\theta$ is the fraction $\frac{\theta}{360^\circ}$ of the circle, so its area is that fraction of $\pi r^2$ .

A = \frac{\theta}{360^\circ} \cdot \pi r^2

Compare this with arc length, $\frac{\theta}{360^\circ} \cdot 2\pi r$ . Both use the same fraction. Arc length takes that fraction of the circumference, and sector area takes it of the area. Area is in square units.

How to Find the Area of a Sector

Find the radius. If you know the diameter, divide it by 2.
Find the central angle $\theta$ .
Write the fraction $\frac{\theta}{360^\circ}$ and simplify it.
Find the circle’s area, $\pi r^2$ , and multiply.
Give the exact answer in terms of $\pi$ , then round if asked. Use square units.

Worked Examples

Example 1: Find the area of a sector

A circle has a radius of 6 cm. Find the area of a sector with a central angle of 100°.

Write the formula: $A = \frac{100}{360} \cdot \pi(6)^2$
Square the radius: $A = \frac{100}{360} \cdot 36\pi$
Simplify the fraction (see fractions): $A = \frac{5}{18} \cdot 36\pi$
Multiply (see multiplying and dividing fractions): $A = 10\pi$
Use a calculator: $A \approx 31.42$

Answer: $10\pi$ cm², or about 31.42 cm²

Example 2: Another sector

Find the area of a 72° sector of a circle with radius 10 m.

Write the formula: $A = \frac{72}{360} \cdot \pi(10)^2$
Square the radius: $A = \frac{72}{360} \cdot 100\pi$
Simplify the fraction: $A = \frac{1}{5} \cdot 100\pi$
Multiply: $A = 20\pi$
Use a calculator: $A \approx 62.83$

Answer: $20\pi$ m², or about 62.83 m²

Example 3: Find the central angle

A sector of a circle with radius 6 has area $15\pi$ . Find its central angle.

Write the formula: $15\pi = \frac{\theta}{360} \cdot \pi(6)^2$
Square the radius: $15\pi = \frac{\theta}{360} \cdot 36\pi$
Divide both sides by $36\pi$ : $\frac{15}{36} = \frac{\theta}{360}$
Multiply both sides by 360: $\theta = \frac{15}{36} \cdot 360$
Multiply: $\theta = 150$

Answer: 150°

Example 4: Find the radius

A 90° sector has area $16\pi$ in². Find the radius.

Write the formula: $16\pi = \frac{90}{360} \cdot \pi r^2$
Simplify the fraction: $16\pi = \frac{1}{4}\pi r^2$
Multiply both sides by 4: $64\pi = \pi r^2$
Divide both sides by $\pi$ : $64 = r^2$
Take the positive square root: $r = 8$

Answer: The radius is 8 in.

Example 5: Word problem with pizza

A 16-inch pizza (16 in across) is cut into 8 equal slices. What is the area of one slice?

Find the radius: $r = 16 \div 2 = 8$
Find the central angle of one slice: $360^\circ \div 8 = 45^\circ$
Write the formula: $A = \frac{45}{360} \cdot \pi(8)^2$
Square the radius: $A = \frac{45}{360} \cdot 64\pi$
Simplify the fraction: $A = \frac{1}{8} \cdot 64\pi$
Multiply: $A = 8\pi$
Use a calculator: $A \approx 25.13$

Answer: Each slice is $8\pi$ in², or about 25.13 in².

Common Mistakes

Mistake	Example of the error	Fix
Using the circumference instead of the area	$\frac{100}{360} \cdot 2\pi(6)$	That is arc length. Sector area uses $\pi r^2$ .
Squaring the diameter	$\pi(16)^2$ for a 16-inch pizza	Use the radius: $\pi(8)^2$ .
Forgetting square units	Area = $10\pi$ cm	Area is in square units: $10\pi$ cm².
Squaring before halving	$\frac{16^2}{2}$ for the radius squared	Halve the diameter first, then square: $8^2 = 64$ .
Using the minor angle for a major sector	Using 80° for the large part	Major sector angle = $360^\circ - 80^\circ = 280^\circ$ .

Related Idea: Area of a Segment

A segment of a circle is the region between a chord and its arc. To find its area, subtract the triangle from the sector.

Circle with center O and radius 6. A 90 degree sector is split into two parts by a chord: a right triangle with legs 6 and 6, shaded blue, and a segment between the chord and the arc, shaded orange. The segment area equals the sector area, 9 pi, minus the triangle area, 18, which is about 10.27. — Segment area = sector area − triangle area.

For a 90° sector with radius 6, find the sector: $\frac{90}{360} \cdot \pi(6)^2 = \frac{1}{4} \cdot 36\pi = 9\pi$
Find the triangle (see area of triangles): $\frac{1}{2} \cdot 6 \cdot 6 = 18$
Subtract: segment $= 9\pi - 18$
Use a calculator: $9\pi - 18 \approx 10.27$

Practice Problems

Give each answer exactly in terms of $\pi$ , then round to the nearest hundredth.

Find the area of a 90° sector of a circle with radius 4 cm.
Find the area of a 40° sector of a circle with radius 9 m.
Find the area of a 216° sector of a circle with diameter 20 ft.
A sector of a circle with radius 6 has area $6\pi$ . Find its central angle.
A 120° sector has area $27\pi$ . Find the radius.
In a circle of radius 3, a minor sector has a central angle of 80°. Find the area of the major sector.
A lawn sprinkler sprays water 10 m and turns through 135°. What area does it water?
A pie with a diameter of 9 in is cut into 6 equal pieces. Find the area of one piece.
Find the area of the segment cut off by a 90° arc in a circle with radius 8.
A circle has radius 12. A sector has a central angle of 30°. Find its arc length and its area.

Show answers

Write the formula: $A = \frac{90}{360} \cdot \pi(4)^2$
Square the radius: $A = \frac{1}{4} \cdot 16\pi$
Multiply: $A = 4\pi$
Use a calculator: $A \approx 12.57$ cm²
Write the formula: $A = \frac{40}{360} \cdot \pi(9)^2$
Square the radius: $A = \frac{1}{9} \cdot 81\pi$
Multiply: $A = 9\pi$
Use a calculator: $A \approx 28.27$ m²
Find the radius: $r = 20 \div 2 = 10$
Write the formula: $A = \frac{216}{360} \cdot \pi(10)^2$
Square the radius: $A = \frac{216}{360} \cdot 100\pi$
Simplify the fraction: $A = \frac{3}{5} \cdot 100\pi$
Multiply: $A = 60\pi$
Use a calculator: $A \approx 188.50$ ft²
Write the formula: $6\pi = \frac{\theta}{360} \cdot \pi(6)^2$
Square the radius: $6\pi = \frac{\theta}{360} \cdot 36\pi$
Divide both sides by $36\pi$ : $\frac{6}{36} = \frac{\theta}{360}$
Multiply both sides by 360: $\theta = \frac{6}{36} \cdot 360$
Multiply: $\theta = 60^\circ$
Write the formula: $27\pi = \frac{120}{360} \cdot \pi r^2$
Simplify the fraction: $27\pi = \frac{1}{3}\pi r^2$
Multiply both sides by 3: $81\pi = \pi r^2$
Divide both sides by $\pi$ : $81 = r^2$
Take the positive square root: $r = 9$
Find the major angle: $360^\circ - 80^\circ = 280^\circ$
Write the formula: $A = \frac{280}{360} \cdot \pi(3)^2$
Square the radius: $A = \frac{280}{360} \cdot 9\pi$
Simplify the fraction: $A = \frac{7}{9} \cdot 9\pi$
Multiply: $A = 7\pi$
Use a calculator: $A \approx 21.99$
Write the formula: $A = \frac{135}{360} \cdot \pi(10)^2$
Square the radius: $A = \frac{135}{360} \cdot 100\pi$
Simplify the fraction: $A = \frac{3}{8} \cdot 100\pi$
Multiply: $A = 37.5\pi$
Use a calculator: $A \approx 117.81$ m²
Find the radius: $r = 9 \div 2 = 4.5$
Find the angle of one piece: $360^\circ \div 6 = 60^\circ$
Write the formula: $A = \frac{60}{360} \cdot \pi(4.5)^2$
Square the radius: $A = \frac{1}{6} \cdot 20.25\pi$
Multiply: $A = 3.375\pi$
Use a calculator: $A \approx 10.60$ in²
Find the sector: $\frac{90}{360} \cdot \pi(8)^2 = \frac{1}{4} \cdot 64\pi = 16\pi$
Find the triangle: $\frac{1}{2} \cdot 8 \cdot 8 = 32$
Subtract: $16\pi - 32$
Use a calculator: $\approx 18.27$
Arc length: $\frac{30}{360} \cdot 2\pi(12) = \frac{1}{12} \cdot 24\pi = 2\pi \approx 6.28$
Area: $\frac{30}{360} \cdot \pi(12)^2 = \frac{1}{12} \cdot 144\pi = 12\pi \approx 37.70$

FAQ

What is the formula for the area of a sector?

$A = \frac{\theta}{360^\circ} \cdot \pi r^2$ , where $\theta$ is the central angle in degrees and $r$ is the radius.

What is the difference between a sector and a segment?

A sector is bounded by two radii and an arc, like a pizza slice. A segment is bounded by a chord and an arc. Segment area is sector area minus the triangle.

How is sector area related to arc length?

Both use the fraction $\frac{\theta}{360^\circ}$ . Arc length takes that fraction of the circumference. Sector area takes it of the area.

How do you find the angle of a sector from its area?

Divide the sector area by the circle’s area to get the fraction, then multiply by 360°. For area $15\pi$ in a circle of area $36\pi$ , the angle is $\frac{15}{36} \cdot 360^\circ = 150^\circ$ .

Summary

A sector is a pie-shaped region between two radii and an arc.
Sector area = $\frac{\theta}{360^\circ} \cdot \pi r^2$ .
Use the radius, simplify the fraction, and give the answer in square units.
Work backward to find the angle or the radius.
Segment area = sector area − triangle area.

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