Lesson Explainer: Areas of Circular Sectors Mathematics

In this explainer, we will learn how to find the area of a circular sector and solve problems that relate this area to the arc length and perimeter of the sector.

Before we determine how to find the area of a circular sector, let’s start by recalling the terminology for the parts of a sector of a circle. First, we recap that an arc of a circle is a section of the circle between two radii. However, given two radii, there are two possible arcs between the two radii. We can see an example of this in the following diagram.

We can see both arcs are a section of the circle between the two given radii. To get around this problem, we denote the larger arc “major” and the smaller as “minor.”

This is equivalent to saying that if the central angle is less than 180 or 𝜋 radians, then we know it is minor and that if it is greater than these values, then it is major. We can then define circular arcs as follows.

Recap: An Arc of a Circle

An arc of a circle is a section of the circumference of a circle between two radii.

Given two radii, we denote the larger of the arcs as the major arc and the smaller of the arcs as the minor arc. The larger arc is the one with the larger central angle.

Finally, if the two arcs are the same length, then we call them semicircular arcs. These occur when the central angle is 180 or 𝜋 rad, or, equivalently, when the radii form a diameter.

Definition: A Sector of a Circle

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

If the central angle is less than 180 or 𝜋 rad, we refer to this as a minor sector. If the central angle is larger than that, we refer to this as a major sector. And if the angle is equal to 180, then our sector is a semicircle.

How To: Calculating the Area of a Circular Sector

We are now ready to find the area of a sector. Let’s start with an example where the central angle is 90.

This is a quarter of the circle, so we can calculate the area of this sector by multiplying the area of the whole circle 𝜋𝑟 by 14. More generally, for a sector with central angle 𝜃 measured in degrees, this is a 𝜃360 portion of the circle and its area can be calculated using the formula 𝜃360𝜋𝑟.

We can do the same in radians. If the central angle of the sector measured in radians is 𝜃, then the sector is a 𝜃2𝜋 portion of the circle. So, the area of this sector is 𝜃2𝜋𝜋𝑟=12𝑟𝜃.

This the gives us the following formulas for finding the areas of circular sectors.

Formulas: Area of a Circular Sector

The area of a sector of radius 𝑟 and central angle 𝜃 measured in degrees is given by 𝐴=𝜃360𝜋𝑟.

The area of a sector of radius 𝑟 and central angle 𝜃 measured in radians is given by 𝐴=12𝑟𝜃.

Let’s see an example of how to apply the formula for the area of a circular sector.

Example 1: Calculating the Area of a Sector given the Measure of Its Angle in Radians

An arc has a measure of 𝜋3 radians and a radius of 5. Give the area of the sector, in terms of 𝜋, in its simplest form.

Answer

Recall that, for a circle of radius 𝑟, if the measure of the central angle of a sector is 𝜃 radians, then the area of the sector is given by 𝐴=12𝑟𝜃.

If we were to sketch the information given, we would get the following:

We have 𝑟=5 and 𝜃=𝜋3, giving us 𝐴=12𝜋35=25𝜋6.

Therefore, the area of this sector is 25𝜋6 area units.

We can also use the formula for the area of a sector to determine the area of more complicated shapes, as we will see in the following example.

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

Find the area of the colored part of the diagram, giving the answer to one decimal place.

Answer

The shaded area in the diagram is the difference in area between two sectors of two circles sharing a center, as shown in the diagram below.

The radius of the inner circle is 7 cm and the radius of the outer circle is 15 cm. We can find the shaded area by finding the area of the sector of the larger circle and then subtracting the area of the sector of the inner circle. Let’s start with the sector of the outer circle.

We recall the area of the sector of a circle of radius 𝑟 with central angle 𝜃 measured in degrees is given by 𝐴=𝜃360𝜋𝑟.

The radius is 15 cm and the central angle is 60. Substituting these into the formula gives outersectorareacm=60360𝜋(15)=75𝜋2.

We can then find the area of the inner circle’s sector.

The radius is 7 cm and the central angle is 60, giving innersectorareacm=60360𝜋(7)=49𝜋6.

Finally, the shaded area is the difference in these values: shadedareacm=75𝜋249𝜋6=88𝜋392.153.

Hence, the area of the shaded region in the diagram is 92.2 cm2 to one decimal place.

In our next example, we will use the radius of a circle and the perimeter of a sector to determine the area of the sector. Before we do this, we recall that, for a circle of radius 𝑟 and an arc whose central angle is 𝜃, the arc length 𝑙 can be found as follows:

  • If 𝜃 is measured in radians, the arc’s length is 𝑙=𝑟 and the area of the sector is 𝐴=12𝑟𝜃; so 𝐴=12𝑟𝜃=12𝑟(𝑟𝜃)=12𝑟𝑙.
  • If 𝜃 is measured in degrees, the arc’s length is 𝑙=2𝜋𝑟𝜃360 and the area of the sector is 𝐴=𝜃360𝜋𝑟; so 𝐴=𝜃360𝜋𝑟=12𝑟2𝜋𝑟𝜃360=12𝑟𝑙.

In both cases, the area of the sector is one-half the product of the radius and the arc length.

Formula: Area of a Circular Sector Using Arc Length

If a sector of a circle of radius 𝑟 has arc length 𝑙, then the area 𝐴 of the sector is given by 𝐴=12𝑟𝑙.

Let’s see how to apply this result in our next example.

Example 3: Finding the Area of a Sector given the Radius of the Circle and the Perimeter of the Sector

The radius of a circle is 10 cm and the perimeter of a sector is 25 cm. Find the area of the sector.

Answer

Since we are not given the central angle of the sector and are instead given the perimeter of the sector, we begin by recalling the following formula for the area of a sector. If a sector of a circle of radius 𝑟 has arc length 𝑙, then the area 𝐴 of the sector is given by 𝐴=12𝑟𝑙.

We are told that 𝑟=10cm, and we can find the value of 𝑙 by using the perimeter. We start by sketching the given information.

The perimeter of the sector is the sum of the lengths of its sides, so we have 25=10+10+𝑙𝑙=5.

Hence, substituting 𝑙=5 and 𝑟=10 into the area formula, we have 𝐴=12×5×10=25.cm

Therefore, the area of this sector is 25 cm2.

In our next example, we will see how we can apply geometric properties along with the formula for the area of a sector of a circle to find areas given in a diagram.

Example 4: Circular Sectors and Areas of Circles

Three congruent circles with a radius of 43 cm are placed touching each other. Find the area of the part between the circles giving the answer to the nearest square centimetre.

Answer

To find the area between the circles, we will start by joining the centers of the circles together via the line segments. The area we need to find is the shaded region in the following diagram. All three circles have a radius of 43 cm, so we can mark the lengths of the following radii.

We can then see this is an equilateral triangle with side length 86 cm. Since this is an equliateral triangle, the internal angles are 60. Instead of finding the shaded region, we can find the area of the triangle and then subtract the areas of the sectors.

The area of the triangle is half the length of the base times the height. To find the height of the triangle, we construct the following right triangle:

The height of the equilateral triangle divides it into two congruent right triangles. The Hypotenuse of one of these right triangles has length 86 cm and the base is half of a side of the equilateral triangle and so has length 862=43cm. We can then use the Pythagorean theorem to find : 86=43+=8643=5547.

Then, by using the area of a triangle, half the length of the base multiplied by the perpendicular height, we have areaofthetrianglecm=12×86×5547=18493.

We recall that the area of a sector of radius 𝑟 and central angle 𝜃 measured in degrees is given by 𝐴=𝜃360𝜋𝑟.

Substituting 𝜃=60 and 𝑟=43cm into the formula for the area of a sector gives 𝐴=60360𝜋(43)=1849𝜋6.

We can then find the area between the circles by subtracting the areas of the three sectors from the area of the triangle. This gives us areacm=1849331849𝜋6=184931849𝜋2298.159.

Hence, the area of the region between the circles in the diagram, to the nearest square centimetre, is 298 cm2.

In our final example, we will see how to apply the formula for the area of the sector in a real-world problem.

Example 5: Solving Problems Involving Sectors

A landscape gardener decides that he wants to design a lawn split into a series of sectors with circular patios laid into the grass, as shown in the given figure. The circular lawn will be split into six equal sectors, each with a radius of eight metres. The lines 𝑂𝐴 and 𝑂𝐵 are both tangents to the circle, and the arc 𝐴𝐵 touches the circle at a single point.

  1. Work out the area of sector 𝑂𝐴𝐵. Give your answer in terms of 𝜋.
  2. The gardener needs to calculate the radius of the circular patio. Using trigonometric ratios, calculate the radius of the patio. Give your answer as a fraction.
  3. Calculate the total area of grass in one sector. Give your answer, in terms of 𝜋, in its simplest form.

Answer

Part 1

We recall that the area of a sector of radius 𝑟 and central angle 𝜃 measured in degrees is given by the following: areaofasector=𝜃360𝜋𝑟.

Substituting 𝑟=8 and 𝜃=60 into this formula gives us areaofthesector=60360𝜋(8)=32𝜋3.

Since the lengths are measured in metres, this sector has area 32𝜋3 square metres.

Part 2

To find the radius of this circle, we start by adding the following lines and points to the diagram:

Since 𝑂𝐴 is a tangent to the circle, it meets the circle at a right angle, meaning that 𝑂𝑃𝐶 is a right triangle. We know the lengths of 𝑃𝐶 and 𝐶𝐷 are 𝑟, since they are radii of the inner circle. We also know that 𝑂𝐷 has a length of 8 because it is a radius of the outer circle. We choose 𝑂𝐷 such that it bisects the angle at 𝑂.

To use trigonometry on triangle 𝑂𝑃𝐶, we need to find the length of 𝑂𝐶: 𝑂𝐷=𝑂𝐶+𝐶𝐷.

We know 𝑂𝐷 has a length of 8 and 𝐶𝐷 has a length of 𝑟, so 8=𝑂𝐶+𝑟𝑂𝐶=8𝑟.

Using the sine ratio, we have sin(30)=𝑃𝐶𝑂𝐶; we can then solve for 𝑟: sin(30)=𝑟8𝑟12=𝑟8𝑟8𝑟=2𝑟𝑟=83.

Hence, the radius of the patio is 83 metres.

Part 3

The area of the grass in one sector will be the area of the patio subtracted from the area of the sector.

The area of a circle of radius 𝑟 is 𝜋𝑟, so the area of the patio is 𝜋83=64𝜋9.squaremetres

In part 1, we showed the area of the sector was 32𝜋3 square metres. Therefore, the area of the grass is 32𝜋364𝜋9=32𝜋9.squaremetres

Let’s finish by recapping some of the important points of this explainer.

Key Points

  • A sector of a circle is a part of a circle enclosed by two radii and an arc between them.
  • The area of a sector of radius 𝑟 and central angle 𝜃 measured in degrees is given by 𝐴=𝜃360𝜋𝑟.
  • The area of a sector of radius 𝑟 and central angle 𝜃 measured in radians is given by 𝐴=12𝑟𝜃.
  • If a sector of a circle of radius 𝑟 has arc length 𝑙, then the area 𝐴 of the sector is given by 𝐴=12𝑟𝑙.

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