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Area of Sectors and Segments of a Circle

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Parts of a circle

A circle is the set of all points equidistant from a given point on a plane. It encloses a measurable area. You can divide circles into sectors or pieces. Pieces cut off by connecting any two points on the circle are segments.

Since both sectors and segments are part of a circle's interior, both have area. We can calculate their area using formulas.

Pies, cakes, pizzas; so many foods we eat neatly lend themselves to mathematics, because they are models of circles.

The area of a circle is always calculated using the known relationship of $\pi$ between a circle's radius, r, (or diameter, d) and its circumference:

A=\pi {r}^{2}

A=\pi (\frac{d}{2}{)}^{2}

When you take any two radii of the circle, the area between the radii is a sector:

Picture Circle A with 1/4th sector formed from Points R and P {radii RA and PA}

A chord is a line created by connecting any two points on the circle, without worrying about the center. A chord creates an area called a segment.

The areas of both segments and sectors can be calculated in square units of whatever linear measurement you are given.

Get free estimates from geometry tutors near you.

Area of a sector

For Pi Day, your Math Club is celebrating with pies. You make or buy an 8" blueberry pie. You can quickly calculate the area of the pie, using either formula:

A=\pi {r}^{2}

A=\pi \times (4in{)}^{2}

A=\pi \times 16i{n}^{2}

A=50.265482 i{n}^{2}

Now try it with the other formula.

A=\pi (\frac{d}{2}{)}^{2}

A=\pi (\frac{8in}{2}{)}^{2}

A=3.14159265\times 16i{n}^{2}

A=50.265482 i{n}^{2}

Buyers of whole pies can be charged your Math Club's usual cost, $0.31 cents a square inch, but what if someone wants only $\frac{1}{4}$ of the pie, or even a $\frac{1}{16}$ of a pie?

To calculate the area of a fractional part of the pie, you can set up a modified version of the area formula for the whole pie:

$A=\times \pi {r}^{2}$
$A=\frac{1}{4}\times \pi \times {r}^{2}$

$A=\frac{1}{16}\times \pi \times {r}^{2}$

But you can also use a general formula, based on the central angle of the slice, represented by n° or $\theta$ :

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

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

Area of a sector example

The Math Club has 9.5" diameter apple pies to sell. You cut them into 45° slices. What is the area of each slice?

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

A=\frac{45°}{360°}\times \pi \times (4.75in{)}^{2}

A=0.125\times 3.14159265\times 22.5625 i{n}^{2}

A=8.860273 i{n}^{2}

Some folks only have pocket change. What is the area of a 10° slice?

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

A=\frac{10°}{360°}\times \pi \times (4.75 in{)}^{2}

A=0.027777\times 3.14159265\times 22.5625 i{n}^{2}

A=1.968894 i{n}^{2}

What would a 10° slice cost, at $0.31 a square inch? Only $0.61. Such a deal!

Area of segment of a circle

To calculate areas of segments, you first have to know the area of the sector. You may think of the sector as a triangle and a segment put together.

To find the area of the segment, you find the area of the sector. Then you find the area of the isosceles triangle at the central angle. Subtract the area of the isosceles triangle from the area of the sector and you have the area of the segment:

A (segment) = A (sector) - A (triangle)

If you know the radius, r, of the circle and you know the central angle, $\theta$ , in radians of the sector that contains the segment, you can use this formula to calculate the area, A, of only the segment:

A=\frac{1}{2}\times {r}^{2}(\theta -\sin\left(\theta \right))

If you know the radius, r, of the circle and you know the central angle, $\theta$ , in degrees of the sector that contains the segment, you can use this formula to calculate the area, A, of only the segment:

A=\frac{1}{2}\times {r}^{2}\times ((\frac{\pi }{180})\theta -\sin\left(\theta \right))

For example, take those 9.5" pies again. A sector (slice) of pie with a central angle of 60° or $\frac{\pi }{3}$ radians for a pie with a radius of 9.5" would plug into the formulas like this:

A=\frac{1}{2}\times {r}^{2}(\theta -\sin\left(\theta \right))

A=\frac{1}{2}\times {9.5}^{2}\times (\frac{\pi }{3}-\sin\left(\frac{\pi }{3}\right))

A=0.5\times 90.25\times 0.18117214

A=8.1753931 sq in

Or:

A=\frac{1}{2}\times {r}^{2}\times ((\frac{\pi }{180})\theta -\sin\left(\theta \right))

A=0.5\times {9.5}^{2}\times ((\frac{\pi }{180})60-\sin\left(60\right))

A=0.5\times 90.25\times (0.01745329251\times 60-0.86602540378)

A=0.5\times 90.25\times 0.1811721468

A=8.1753931 sq in

You can be as precise as you wish (and as precise as $\pi$ will allow) in your calculations. Taking the final answer to the seventh decimal place is generally not necessary, but we wanted you to see that both methods return the same value to a very high degree of precision (100-millionths).