Video: Areas of Circular Segments

In this video, we will learn how to find the area of a circular segment.

15:14

Video Transcript

In this video, we will learn how to find the area of a circular segment. We will begin by looking at the definition of a circular segment. We will then identify formulas that can be used to calculate the area of a circular segment and then use these to solve some problems.

What is the definition of a circular segment? Is it (A) a region of a circle bounded by an arc and a chord passing through the end points of the arc? (B) A region of a circle bounded by two radii and an arc. (C) A region of a circle bounded by a chord and a central angle subtended by that arc. (D) A region of a circle bounded between two chords and two arcs. Or (E) an arc which is half of the circumference.

The circle shown has been split into two segments. The smaller part of the circle is known as the minor segment. The larger part is known as the major segment. Both of these segments are bounded by an arc. They are also bounded by a common chord. This means that the correct definition is option (A). A circular segment is a region of a circle bounded by an arc and a chord passing through the end points of the arc. In our diagram, we have a minor segment in orange and a major segment in pink.

We will now look at the formulas that can be used to calculate the area of a segment.

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.

We will now use these to find the area of a segment given different properties of a circle.

The area of a circle is 227 square centimeters and the central angle of a segment is 120 degrees. Find the area of the segment, giving the answer to two decimal places.

We’re told in the question that the central angle of a segment is 120 degrees. And we need to calculate the area of this segment. When the angle of a segment is given in degrees, we can calculate the angle of this segment by subtracting the area of the triangle from the area of the sector. The area of the sector is equal to 𝜃 over 360 multiplied by 𝜋𝑟 squared. The area of the triangle is equal to a half 𝑟 squared multiplied by sin 𝜃. We are told in the question that the area of the circle is equal to 227 square centimeters. This means that 𝜋𝑟 squared equals 227. Dividing both sides of this equation by 𝜋 gives us 𝑟 squared is equal to 227 over 𝜋.

We can now substitute these into both of our formulas. The area of the sector is equal to 120 over 360 multiplied by 227. This can be simplified to one-third multiplied by 227 or 227 over three. The area of the triangle can be calculated by multiplying a half by 227 over 𝜋 by sin of 120 degrees. Sin of 120 degrees is equal to root three over two. The area of the segment can therefore be calculated by subtracting a half multiplied by 227 over 𝜋 multiplied by root three over two from 227 over three. Typing this into the calculator gives us 44.37875 and so on. As we need to round our answer to two decimal places, the key or deciding number is the eight. This means that we round up to 44.38. The area of the segment is 44.38 square centimeters.

As this is the area of the minor segment, we could calculate the area of the major segment by subtracting this answer from 227 square centimeters.

We will now look at another question where we are given the radius and the chord.

A circle has a radius of 10 centimeters. A chord of length 14 centimeters is drawn. Find the area of the major segment, giving the answer to the nearest square centimeter.

We are told that the circle has a radius of 10 centimeters. A chord of length 14 centimeters is drawn on the circle. If we let the two ends of the chord be points 𝐴 and 𝐵 and the center point 𝑂, then the area of the minor segment is equal to the area of the sector minus the area of the triangle. In order to calculate both of these, we firstly need to work out the central angle 𝜃. This can be done in either radians or degrees. In this question, we will use radians. So, it is important that our calculator is in the correct mode. The area of a sector, when 𝜃 is in radians, is equal to a half 𝑟 squared 𝜃. And the area of a triangle is equal to a half 𝑟 squared sin 𝜃. This can be simplified by factoring, giving us the area of the segment equal to a half 𝑟 squared multiplied by 𝜃 minus sin 𝜃.

We can now calculate the angle 𝜃 by using right-angle trigonometry or the cosine rule. In order to calculate the angle in any triangle using the cosine rule, we use the following formula. Cos of 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared divided by two 𝑏𝑐, where 𝑎, 𝑏, and 𝑐 are the three lengths of the triangle and 𝐴 is the one opposite the angle we’re trying to work out. Substituting in our values gives us cos of 𝜃 equals 10 squared plus 10 squared minus 14 squared over two multiplied by 10 multiplied by 10. This simplifies to cos of 𝜃 equals one fiftieth. Ensuring that our calculator is in radian mode, 𝜃 is equal to the inverse cos of one fiftieth. This is equal to 1.55079 and so on radians.

We can now substitute this value into our formula for the area of a segment. The area of the minor segment is equal to 27.5497 and so on. We have been asked to calculate the area of the major segment. This is the area of the whole circle minus the area of the minor segment. The area of a circle is equal to 𝜋𝑟 squared. As our radius is equal to 10 centimeters, the area is equal to 100𝜋. We need to subtract 27.5497 and so on from this. This is equal to 286.6095 and so on. We’re asked to round our answer to the nearest square centimeter. The deciding number is the six in the tenths column. So, we round up to 287 square centimeters. This is the area of the major segment in the circle.

We will now summarize the key points from this video. A segment is a region bounded by an arc and a chord passing through the end points of the arc. Drawing a chord on any circle splits it into two segments, a minor segment and a major segment. As suggested by the name, the major segment is the larger of the two parts. The area of the minor segment, shown in orange on the diagram, is equal to the area of the pink sector minus the area of the blue triangle. In order to calculate these areas, we need the angle 𝜃 in either radians or degrees and the length of the radius 𝑟.

When dealing in degrees, we tend to calculate the two areas separately. The area of the sector is equal to 𝜃 over 360 multiplied by 𝜋𝑟 squared. And the area of the triangle is equal to a half 𝑟 squared multiplied by sin 𝜃. When dealing with radians, we can simplify by factoring. The area of the segment when 𝜃 is in radians is a half 𝑟 squared multiplied by 𝜃 minus sin 𝜃. In any question, we can convert from one angle measurement to the other using the fact that 180 degrees is equal to 𝜋 radians and, therefore, 360 degrees is equal to two 𝜋 radians. In some questions, we might need to use trigonometry to find out the angle 𝜃 first.

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