### Video Transcript

๐ด๐ต๐ถ is a triangle where the measure of angle ๐ด is 152 degrees and ๐ต๐ถ is 11 centimeters. Find the area of the circumcircle giving your answer to the nearest square centimeter.

So weโve 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 triangleโs vertices. To calculate the area of any circle, we use the formula ๐๐ squared. So in order to answer this question, weโll 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 ๐ต and ๐ถ, which will be opposite the third angle of the triangle, angle ๐ด. An alternative way of denoting the side connecting vertices ๐ต and ๐ถ is as lowercase ๐ because itโs opposite the angle uppercase ๐ด. As the information weโre given consists of an angle and the length of its opposite side in a non-right triangle, this suggests that weโre 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 over sin ๐ด is equal to ๐ over sin ๐ต which is equal to ๐ over sin ๐ถ. But there is also a connection between the law of sines ratio and the radius of the triangleโs circumcircle. In fact, this ratio is always equal to twice the radius of the circumcircle. We can calculate this ratio because weโve 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 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 is equal to 11 over two sin 152 degrees. Evaluating this on our calculators, making sure theyโre 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.

Itโs best if we can keep the unrounded value on our calculator display and then square it and multiply by ๐ to avoid introducing any rounding errors. Doing so, it gives 431.178. Weโre 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.