Video Transcript
The area of a circular segment is 34 square centimeters, and the central angle is 75 degrees. Find the radius of the circle, giving the answer to the nearest centimeter.
Let’s begin by sketching this circular segment. Here we have a circle. A sector of this circle will be enclosed by an arc and two radii of the circle. The central angle of the sector is the angle at the center of the circle, the angle between the two radii, which we’re told in the question is 75 degrees. Now this isn’t a segment; it is a sector.
To create a segment, we draw in a chord connecting the endpoints of the two radii. The segment is this part shaded in pink, which is bounded by an arc and a chord which passes through the endpoints of the arc. We’re told that the area of this circular segment is 34 square centimeters. And we want to determine the radius of the circle. Now, in general, the area of a circular segment can be found as the difference between the area of a circular sector, that’s this area outlined in orange, and the area of the triangle formed by the two radii and the chord. That’s the shape outlined in green.
There are formulae that we can use to calculate the areas of each of these shapes. The area of a sector with a central angle of 𝜃 degrees and a radius of 𝑟 units is 𝜋𝑟 squared 𝜃 over 360. And using the trigonometric formula for the area of a triangle, the area of this green triangle is a half 𝑟 squared sin 𝜃. A factored form of this expression is 𝑟 squared over two multiplied by 𝜋𝜃 over 180 minus sin of 𝜃. And we can, in fact, learn this as a general formula.
Now, we know the value of 𝜃, the central angle of the sector, is 75 degrees. And we also know the area of the segment. We’re told in the question that it is 34 square centimeters. So, substituting these values, we have an equation. 34 is equal to 𝑟 squared over two multiplied by 75𝜋 over 180 minus sin of 75 degrees. And we can now solve this equation to determine the value of 𝑟.
We’ll begin by multiplying both sides of the equation by two to give 68 is equal to 𝑟 squared multiplied by 75𝜋 over 180 minus sin of 75 degrees. Now, we can evaluate this constant of 75𝜋 over 180 minus sin of 75 degrees, and it gives 0.343 continuing. But we’ll keep that exact value on our calculator display in order to prevent any rounding errors.
We can then divide both sides of the equation by this value to give 𝑟 squared equals 68 over 0.343 continuing, which is 198.209 continuing. To find 𝑟, we then take the square root of this value. And remember, we’ll keep exact values on our calculator display throughout so that our final answer is as accurate as possible. This gives 14.078 continuing.
The question specifies that we should give our answer to the nearest centimeter. So we’ll round this value to the nearest integer, which is 14. And we include the units. We found then that the radius of this circle to the nearest centimeter is 14 centimeters.
