Video: Finding the Area of a Segment given the Area of the Circle and the Segment's Central Angle

The area of a circle is 227 cmΒ² and the central angle of a segment is 120Β°. Find the area of the segment giving the answer to two decimal places.

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Video Transcript

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 πœƒ over 360 multiplied by πœ‹π‘Ÿ squared. The area of the triangle is equal to a half π‘Ÿ squared multiplied by sin πœƒ. We are told in the question that the area of the circle is equal to 227 square centimeters. This means that πœ‹π‘Ÿ squared equals 227. Dividing both sides of this equation by πœ‹ gives us π‘Ÿ squared is equal to 227 over πœ‹.

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œ‹ 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œ‹ 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.

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