Video: Finding the Area of a Circular Segment

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

03:47

Video Transcript

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 𝐡 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 πœƒ. 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 πœƒ is in radians, is equal to a half π‘Ÿ squared πœƒ. And the area of a triangle is equal to a half π‘Ÿ squared sin πœƒ. This can be simplified by factoring, giving us the area of the segment equal to a half π‘Ÿ squared multiplied by πœƒ minus sin πœƒ.

We can now calculate the angle πœƒ 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 π‘Ž squared divided by two 𝑏𝑐, where π‘Ž, 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 πœƒ equals 10 squared plus 10 squared minus 14 squared over two multiplied by 10 multiplied by 10. This simplifies to cos of πœƒ equals one fiftieth. Ensuring that our calculator is in radian mode, πœƒ 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 πœ‹π‘Ÿ squared. As our radius is equal to 10 centimeters, the area is equal to 100πœ‹. 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.

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