Question Video: Using the Law of Sines and Trigonometric Formula for Area of Triangles to Calculate the Areas of Circular Segments Mathematics

Video Transcript

Find the area of the green part of the diagram, given that the measure of angle 𝐶𝐴𝐵 is 77 degrees, the measure of angle 𝐵𝐶𝐴 is 57 degrees, and 𝐶𝐵 is 19 centimeters. Give the answer to the nearest square centimeter.

Let’s begin by adding the information given in the question to the diagram. Firstly, the measure of angle 𝐶𝐴𝐵 is 77 degrees. Then, the measure of angle 𝐵𝐶𝐴 is 57 degrees. And finally, the length of the side 𝐶𝐵 is 19 centimeters. We may also find it useful to use the lowercase letters 𝑎, 𝑏, and 𝑐 to represent the sides’ opposite angles 𝐴, 𝐵, and 𝐶, respectively.

We’re looking to find the area of the green part of the diagram, or the shaded area, which we can find by finding the area of the circle and then subtracting the area of triangle 𝐴𝐵𝐶. We’ll need to think individually about how we can find each of these areas.

Let’s think about triangle 𝐴𝐵𝐶 first. This is a nonright triangle, so let’s recall the trigonometric formula for the area of a triangle. It is a half 𝑎𝑏 sin 𝑐, where 𝑎 and 𝑏 represent two of the triangle side lengths and 𝑐 represents the measure of their included angle. That’s the angle between the two known sides. In our triangle, we only know one of the side lengths, side 𝑎. So, let’s think about how we could calculate another side length.

We know the length of one side and the measure of its opposite angle, which means we can calculate the length of either of the other two sides if we know the measure of their opposite angle by applying the law of sines. This tells us that in a triangle the ratio between any side length and the sine of its opposite angle is constant, which we can express as 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶. We could calculate the length of side 𝑐 as we know the measure of its opposite angle. Or as we know two angles in the triangle, we can easily calculate the measure of the third angle, angle 𝐵, and then calculate the length of side 𝑏. Let’s do that.

The measure of angle 𝐵 then is 180 degrees minus 77 degrees minus 57 degrees, which is 46 degrees. We’re now able to substitute the known lengths and angles into the law of sines. So, we have 𝑎 over the sin of angle 𝐴. That’s 19 over sin of 77 degrees. This is equal to 𝑏 over the sin of angle 𝐵. That’s 𝑏 over sin of 46 degrees. And now, we have an equation we can solve to find the length of side 𝑏. Multiplying both sides of the equation by sin of 46 degrees gives 𝑏 equals 19 sin 46 degrees over sin of 77 degrees. Evaluating this on a calculator, which must be in degree mode, we find that 𝑏 is equal to 14.026 continuing.

We’re now in a position to find the area of this triangle as we know the lengths of two sides and the measure of their included angle. Remember, we could’ve found the length of side 𝑐 earlier on, but we would still have need to have found the measure of angle 𝐵 as this is the included angle between sides 𝑎 and 𝑐. So, the amount of work we’d need to do is the same. Using sides 𝑎 and 𝑏 then and the measure of angle 𝐶, we have that the area of this triangle is equal to a half multiplied by 19 multiplied by 14.026 — and we’ll keep that value on our calculator display so that it’s accurate — multiplied by sin of 57 degrees, which is equal to 111.758 continuing.

So, we’ve found the area of triangle 𝐴𝐵𝐶, and now we need to consider how to find the area of the circle. We can find the area of any circle using the formula 𝜋𝑟 squared. So, we need to consider how we can calculate the radius of this circle. Well, this is, in fact, the circumcircle of triangle 𝐴𝐵𝐶 as all three of the triangles’ vertices lie on the circumference of the circle. We should recall that there is, in fact, a connection between the law of sines ratio and the radius of a triangle circumcircle. The law of sines ratio is, in fact, equal to twice the radius of the circumcircle.

We’ve already written this ratio down using the length of side 𝑎 and the measure of its opposite angle when we calculated the length of side 𝑏. So, we can say that twice the radius of the circumcircle is equal to 19 over sin of 77 degrees. It then follows that the radius is equal to 19 over two sin 77 degrees, which is equal to 9.749 continuing. The area of the circle then is equal to 𝜋 multiplied by this value squared. And again, we’ll keep the exact value on our calculator display to prevent any rounding errors. This gives 298.640 continuing.

We’ve now found both the area of the circle and the area of the triangle. So, we’re finally able to calculate the area of the green part of the diagram. Using values as exact as possible, we have that the shaded area is equal to 298.640 minus 111.758. That gives 186.8 continuing. And then, we need to round our answer to the nearest square centimeter.

Using the law of sines then and recalling its connection with the triangle’s circumcircle, we found that the area of the green part of the diagram to the nearest square centimeter is 187 centimeters squared.