Question Video: Finding the Area of a Circumcircle Given the Measure of an Angle and the Length of the Opposite Side Mathematics

๐ด๐ต๐ถ is a triangle where ๐‘šโˆ ๐ด = 152ยฐ, and ๐ต๐ถ = 11 cm. Find the area of the circumcircle giving the answer to the nearest square centimeter.

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Video Transcript

๐ด๐ต๐ถ is a triangle where the measure of angle ๐ด is 152 degrees and ๐ต๐ถ is 11 centimeters. Find the area of the circumcircle giving your answer to the nearest square centimeter.

So weโ€™ve been given some information about a triangle and asked to find the area of its circumcircle. We recall first that the circumcircle of a triangle is the circle passing through all three of the triangleโ€™s vertices. To calculate the area of any circle, we use the formula ๐œ‹๐‘Ÿ squared. So in order to answer this question, weโ€™ll need to work out the radius of this circle.

The information given about this triangle consists of the measure of one angle and the length of its opposite side. We know this because the length given is for the side connecting vertices ๐ต and ๐ถ, which will be opposite the third angle of the triangle, angle ๐ด. An alternative way of denoting the side connecting vertices ๐ต and ๐ถ is as lowercase ๐‘Ž because itโ€™s opposite the angle uppercase ๐ด. As the information weโ€™re given consists of an angle and the length of its opposite side in a non-right triangle, this suggests that weโ€™re going to be using the law of sines.

The law of sines tells us that in any triangle the ratio between the length of each side and the sine of its opposite angle is constant, which we can express as ๐‘Ž over sin ๐ด is equal to ๐‘ over sin ๐ต which is equal to ๐‘ over sin ๐ถ. But there is also a connection between the law of sines ratio and the radius of the triangleโ€™s circumcircle. In fact, this ratio is always equal to twice the radius of the circumcircle. We can calculate this ratio because weโ€™ve already established we have the length of a side and the measure of its opposite angle.

So substituting 11 for a side length and 152 degrees for its opposite angle, we have that two ๐‘Ÿ is equal to 11 over sin of 152 degrees. We can then solve this equation to calculate the radius of the circumcircle. Dividing both sides by two, we have ๐‘Ÿ is equal to 11 over two sin 152 degrees. Evaluating this on our calculators, making sure theyโ€™re in degree mode, gives 11.715. Having calculated the radius of the circumcircle, we now need to calculate its area, so we substitute this value of 11.715 into the formula for the area of a circle.

Itโ€™s best if we can keep the unrounded value on our calculator display and then square it and multiply by ๐œ‹ to avoid introducing any rounding errors. Doing so, it gives 431.178. Weโ€™re asked to give our answer to the nearest square centimeter, so we round this value to the nearest integer and include the units, which are centimeters squared.

So by recalling the connection between the radius of the circumcircle and the law of sines ratio, we found the radius of the circumcircle and, hence, calculated its area, which to the nearest square centimeter is 431 centimeters squared.

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