In this explainer, we will learn how to find the areas of circular sectors and segments when the angles are given in radians.

Before we discuss the areas of circular sectors and segments, we can start by recalling a very similar result involving the length of the arc of a circle. We recall that if an arc subtends an angle of rad in a circle of radius , then the length of the arc is given by

We proved this result by considering the proportion of the entire circumference of the circle as the arc; we can follow the same process by considering the entire area of the circle to find a formula for the area of a portion or a sector of a circle.

Formally, a sector of a circle is a part of a circle enclosed by two radii and an arc between them. There are two sectors of a circle between two radii of a circle, so we refer to the sectors as major, minor, or semicircular based on the arc or the angle subtended by the arc between the radii.

We call the angle at the center of the circle the angle subtended by the arc or sector. The sector that subtends a smaller angle is called the minor sector and the sector that subtends a larger angle is called the major sector. The smaller angle will be less than rad and the larger angle will be greater than rad. If both sectors subtend the same angle, then we call them semicircular arcs and the angles will both be equal to rad.

We can define this formally as follows.

### Definition: A Sector of a Circle

A **sector** of a circle is a part of a circle enclosed by two radii and the arc between them. We name the sectors based
on the type of arc between the radii.

If the sector subtends an angle less than rad, we refer
to this as a **minor sector**, if the sector subtends an angle greater than
rad, we refer to this as a **major sector**, and if
the sector subtends an angle equal to rad, then the
sector is a **semicircle**.

We want to determine the area of a sector of a circle, and the best way to do this is by considering an example.

Consider a sector of a circle of radius that subtends an angle of rad.

This is a quarter of the circle, so we can calculate the area of this sector by multiplying the area of the whole circle by . The reason this works is that the whole circle subtends an angle of , so we can determine the proportion of the circle that is part of the sector by dividing the angles. We have

More generally, a sector that subtends an angle of rad will be a proportion of of the circle. So, the area of this sector is

This then gives us the following formula for finding the area of circular sectors.

### Formula: The Area of a Circular Sector

The area of a sector of a circle of radius that subtends an angle of rad is given by

Letβs see an example of using this formula to determine the area of a sector of a circle from the measure of the arc it subtends and the radius of the circle.

### Example 1: Finding the Area of a Sector given Its Radius and Angle

An arc has a measure of radians and a radius of 5. Give the area of the sector, in terms of , in its simplest form.

### Answer

We begin by recalling that the area of a sector of a circle of radius that subtends an angle of rad is given by

In this question, we are told that an arc has a measure of radians and a radius of 5; this is equivalent to saying that the sector subtends an angle of rad and the radius is 5. Substituting and into the formula yields

In our next example, we will use the area of a sector and the central angle to determine the radius of the circle.

### Example 2: Finding the Radius of a Sector given Its Area and Angle

The area of a circular sector is 1βββ790 cm^{2} and the central angle
is 1.5 rad. Find the radius of the circle giving the answer to the
nearest centimetre.

### Answer

We begin by recalling that the area of a sector of a circle of radius with a central angle of rad is given by

We are told that the area of the sector is 1βββ790 cm^{2} and that the
central angle is 1.5 rad. Substituting and into the equation gives

We note that , so we have

Multiplying both sides of the equation by yields

We now take the square roots of both sides of the equation, where we note that is nonnegative since it is a length:

Rounding this value to the nearest centimetre gives us a radius of approximately 49 cm.

Now, we have seen some examples of how we can find the areas of circular sectors; letβs look at how we can find the areas of circular segments that are regions of a circle enclosed by an arc and a chord.

We can use the formula for the area of a sector combined with the formula for the area of a triangle to determine the area of a circular segment. Letβs start by adding the radii of length to the diagram and letβs say that the arc subtends an angle of rad.

We can note that the shaded region is sector with removed. We can determine the areas of these two shapes separately.

First, sector has radius and it subtends an angle of rad, so its area is . Second, has area . Therefore, if we call the area of the shaded region , we have

Taking out the shared factor of gives

We have shown the following result.

### Formula: The Area of a Circular Segment

The area of a segment of a circle of radius that subtends an angle of rad is given by

Letβs now see an example of using this formula to determine the area of a region in a circle.

### Example 3: Finding the Area of a Shaded Region in a Circle

Find the area of the shaded region of the circle given in the diagram. Give your answer to one decimal place.

### Answer

We begin by recalling that the area of a segment of a circle of radius that subtends an angle of rad is given by . We can use this to determine the area of the unshaded region in the circle. To find the angle subtended by this sector, we recall that the sum of angles around a point is . Thus, this sector subtends an angle of .

Substituting and into the formula for the segment area gives us

This is the area of the unshaded region. We need to subtract this value from the area of the whole circle to find the area of the shaded region. The area of a circle of radius is given by , so this circle has an area of .

Therefore, the area of the shaded region is given by

We round this value to one decimal place to get that the area of the shaded region is approximately
192.9 cm^{2}.

In our next example, we will find the area of a circular segment using its radius and chord length.

### Example 4: Finding the Area of a Segment given Its Radius and Chord Length

Find the shaded area, giving your answer to three decimal places.

### Answer

We are asked to find the area of a circular segment; we can do this by recalling that the area of a segment of a circle of radius that subtends an angle of rad is given by

We are given that the circle has a radius of 5 cm; however, we do not know . We know all three side lengths of since is a radius and therefore must have a length of 5 cm. We can find by using the cosine rule:

Substituting , , , and into the cosine rule yields

We can simplify to get

We can then rearrange as shown:

We now take the inverse cosine of both sides of the equation. We note that is less than rad, so we only need to consider the solution in the second quadrant:

We can now substitute this exact value for into the formula for the segment area along with . We obtain

To three decimal places, we have that the area of the shaded region is
11.182 cm^{2}.

In our next example, we will use the area and radius of a sector to determine its perimeter.

### Example 5: Solving a Word Problem Involving the Area and Perimeter of a Segment

A circular sector has an area of 108 cm^{2} and radius
6 cm. Find the perimeter of the sector.

### Answer

We are given the area and radius of a sector and are asked to determine its perimeter. We can recall that the area of a sector of a circle of radius that subtends an angle of rad is given by . We are told that and , so we can substitute these values into the formula to get

We can then solve the equation for :

We can use this to sketch the major sector.

We can then recall that the length of an arc of a circle that subtends an angle of rad is given by . Therefore, we can substitute and into the formula to find the length of the arc of this sector. We get

The perimeter of the sector is equal to the sum of the two radii and the arc length. Hence, we get

In our final example, we will determine the area of tiling required for a sector-shaped pool.

### Example 6: Solving a Real-World Problem Using the Area of a Segment

A pool is in the shape of a circular sector and the tiles surrounding the front of the pool form a triangle as shown.

Calculate the area of the tiles needed to tile the front of the pool to one decimal place.

### Answer

We want to determine the area of the unshaded region in the diagram. We can find this area by subtracting the area of the sector from the area of the triangle.

We can recall that the area of a sector of a circle of radius that subtends an angle of rad is given by . We can see that ; however, we are not given the value of . Since we have all of the side lengths of the triangle, we can determine this angle by using the cosine rule:

We substitute , , , and into the cosine rule to get

We can evaluate this, giving us

Rearranging the equation yields

We can then solve for by taking the inverse cosine of both sides of the equation and noting that is acute:

We can now find the area of the triangle and sector separately.

First, the area of a triangle is given by the formula , where and are side lengths and is the included angle. We have , , and . Substituting these values into the formula gives

It is important to use the exact values since we do not want to round until the end of the calculation.

The area of a sector of radius subtended by an angle of rad is given by . We have and . Substituting these values into the formula gives

We can now find the difference in these areas to calculate the area of the tiles needed: where we note that it is important to use the exact values in this calculation.

We can now round this to one decimal place to get 61.7 m^{2}.

Letβs finish by recapping some of the important points from this explainer.

### Key Points

- The area of a sector of a circle of radius that subtends an angle of rad is given by .
- The area of a segment of a circle of radius that subtends an angle of rad is given by .