Question Video: Identifying the Area of a Circular Segment given Its Radius and Central Angle Mathematics

Video Transcript

Which formula can be used to find the area of a circular segment, given radius 𝑟 and a central angle 𝜃?

We recall that any circle can be split into a minor and major segment as shown. We’re also told that the radius of the circle is 𝑟 and the central angle is 𝜃. If we label the two points at the end of the chord 𝐴𝐵 and the center of the circle 𝑂, then the area of the segment will be equal to the area of the sector 𝐴𝑂𝐵 minus the area of triangle 𝐴𝑂𝐵. It is important to note at this point that our angle 𝜃 might be given in degrees or in radians. 180 degrees is equal to 𝜋 radians. We know that a circle has a total of 360 degrees, which means it will have a total of two 𝜋 radians. As a result, the area of a circular segment can be calculated using two linked formulas, one for degrees and one when the angle is in radians.

When our angle was measured in degrees, the area of a sector is equal to 𝜃 out of 360 multiplied by 𝜋𝑟 squared. As already mentioned, 360 degrees is equal to two 𝜋 radians. This means that the area of a sector when 𝜃 is in radians is 𝜃 over two 𝜋 multiplied by 𝜋𝑟 squared. In this case, the 𝜋s cancel. We are left with 𝜃 over two multiplied by 𝑟 squared, which is often written as a half 𝑟 squared 𝜃.

As we’ve worked out a formula for the area of a sector in degrees and radians, we will now look at the area of a triangle. The area of any triangle can be calculated using the formula a half of 𝑎𝑏 multiplied by sin 𝐶. In our diagram, we can see that the lengths 𝑎 and 𝑏 are both equal to the radius or 𝑟. The angle 𝐶 is equal to 𝜃. Therefore, the area of a triangle inside a circle can be calculated using the formula half 𝑟 squared multiplied by sin 𝜃. We will now clear some space to work out the formula that can be used to find the area of a circular segment.

Let’s consider when 𝜃 is measured in radians first. The area of the sector is a half 𝑟 squared 𝜃, and the area of the triangle is a half 𝑟 squared sin 𝜃. We can factor out a half 𝑟 squared as this is common in both terms. Inside the parentheses or bracket, we’re left with 𝜃 minus sin 𝜃. When the central angle 𝜃 is given in radians, then the area of the circular segment can be calculated by multiplying a half 𝑟 squared by 𝜃 minus sin 𝜃. If the central angle is given in degrees, then our formula is equal to 𝜃 over 360 multiplied by 𝜋𝑟 squared minus a half 𝑟 squared sin 𝜃.

Whilst the 𝑟 squared is common in both terms, we tend not to factor it out here but instead calculate the area of the sector and area of triangle separately. We then subtract our two answers to calculate the area of the circular segment. Either one of these formulas can be used depending on the context of the question.