Math Core

How much pizza is in one slice? How much lawn does a rotating sprinkler water? Both questions ask for the area of a sector, a wedge of a circle. The idea is the same one you used for arc length: a sector is a fraction of the whole circle, so its area is that same fraction of the circle's area.

Area of a circle

Start with the whole circle. A circle with radius $r$ has area

$A = \pi r^2.$

One way to see this: cut a circle into many thin wedges and rearrange them, alternating tip up and tip down. The result looks almost like a parallelogram whose height is $r$ and whose base is half the circumference, $\pi r$ . Its area is $\pi r \cdot r = \pi r^2$ , and the more wedges you use, the closer the shape gets to a true parallelogram.

Sectors

Definition

Sector

A sector of a circle is the region bounded by two radii and the arc between them. Its central angle is the angle between the two radii.

A 90° sector of a circle with radius 4 is one quarter of the circle.

The sector in the figure has a central angle of $90^\circ$ , which is $\dfrac{90}{360} = \dfrac{1}{4}$ of the full turn. So its area is $\dfrac{1}{4}$ of the circle's area: $\dfrac{1}{4} \cdot \pi(4)^2 = 4\pi$ .

Area of a sector

In a circle of radius $r$ , a sector with central angle $\theta^\circ$ has area

$A = \dfrac{\theta}{360} \cdot \pi r^2.$

If $\theta$ is in radians, the fraction of the circle is $\dfrac{\theta}{2\pi}$ , so

$A = \dfrac{\theta}{2\pi} \cdot \pi r^2 = \dfrac{1}{2} r^2 \theta.$

Worked example: A sector in degrees

Find the area of a sector with central angle $80^\circ$ in a circle of radius $9$ cm.

$A = \dfrac{80}{360} \cdot \pi(9)^2 = \dfrac{2}{9} \cdot 81\pi = 18\pi \text{ cm}^2 \approx 56.5 \text{ cm}^2.$

Worked example: A sector in radians

Find the area of a sector with central angle $\dfrac{\pi}{3}$ radians in a circle of radius $6$ .

$A = \dfrac{1}{2} r^2 \theta = \dfrac{1}{2}(36)\left(\dfrac{\pi}{3}\right) = 6\pi.$

Check with degrees: $\dfrac{\pi}{3}$ radians is $60^\circ$ , and $\dfrac{60}{360} \cdot 36\pi = 6\pi$ .

Worked example: Finding the angle

A sector of a circle with radius $10$ has area $35\pi$ . Find its central angle in degrees.

$\dfrac{\theta}{360} \cdot 100\pi = 35\pi \quad\Longrightarrow\quad \dfrac{\theta}{360} = \dfrac{35}{100} \quad\Longrightarrow\quad \theta = 0.35 \cdot 360 = 126^\circ.$

Common mistake

Square the radius before you do anything else, and don't use the diameter by mistake. A pizza " $14$ inches across" has radius $7$ , so the whole pizza has area $49\pi$ , not $196\pi$ .

Segments of a circle

The region between a chord and its arc is called a segment of the circle. It is what's left of a sector after you remove the triangle formed by the two radii and the chord.

The segment is the region between the chord and the arc: the sector minus the right triangle.

Area of a segment

$\text{area of segment} = \text{area of sector} - \text{area of triangle}$

Worked example: A segment with a right angle

Find the area of the segment cut off by a $90^\circ$ central angle in a circle of radius $4$ .

The sector has area $\dfrac{1}{4}\pi(4)^2 = 4\pi$ . The triangle is a right triangle with legs $4$ and $4$ , so its area is $\dfrac{1}{2}(4)(4) = 8$ .

The segment has area $4\pi - 8 \approx 4.57$ .

Worked example: A segment with a 60° angle

Find the area of the segment cut off by a $60^\circ$ central angle in a circle of radius $6$ .

The sector has area $\dfrac{60}{360}\pi(6)^2 = 6\pi$ . The triangle has two sides of length $6$ and a $60^\circ$ angle between them, so it is equilateral with side $6$ . Its area is $\dfrac{\sqrt{3}}{4}(6)^2 = 9\sqrt{3}$ .

The segment has area $6\pi - 9\sqrt{3} \approx 3.26$ .

Tip

If you know a sector's arc length $s$ and radius $r$ , its area is $A = \dfrac{1}{2}rs$ , just like a triangle with base $s$ and height $r$ . This comes from combining $s = r\theta$ with $A = \dfrac{1}{2}r^2\theta$ .

Practice

Practice 1

Find the area of a $90^\circ$ sector of a circle with radius $6$ . Give your answer in terms of $\pi$ (type pi).