The radius of a circle is 40 centimeters and the arc length of a segment is 18 centimeters. Find the area of the segment giving the answer to two decimal places.
Here’s a diagram of the sector of our circle. We know its radius is 40 centimeters and its arc length is 18 centimeters. Now, we’re trying to find the area of the segment. That’s this bit shaded. And so, we recall that, to find the area of the segment, we find the area of the whole sector and then subtract the area of this triangle. The problem is, to find the area of both the sector and the triangle, we need to calculate the size of the angle 𝜃. And so to do so, we’re going to begin by using the information that the arc length of the segment is 18 centimeters.
Now, if we’re working with degrees, the formula we use to find the arc length is 𝜋 times diameter multiplied by 𝜃 over 360. Essentially, we find a proportion of the full circumference of the circle. When we’re working with radians, though, things are much simpler. The formula we use is 𝑟𝜃. We multiply the length of the radius by the angle 𝜃 in radians. We know that the arc length of our segment is 18 centimeters and the radius of the circle is 40. So, substituting what we know into our formula, and we get 18 equals 40 times 𝜃. And then we divide both sides by 40. So, 𝜃 is 18 over 40, or nine twentieths, radians.
And now we know the angle 𝜃 in radians. We’re ready to calculate the area of the sector. When 𝜃 is measured in radians, the area is a half 𝑟 squared 𝜃. In this case then, the area of our sector is a half times 40 squared times nine twentieths. That’s 360 square centimeters. But what about the area of the triangle? Well, we’re going to use the trigonometric formula a half 𝑎𝑏 sin 𝑐, where 𝑐 is the vertex that sits at the center of our circle. The area of our triangle is a half times 40 times 40 times sin of nine twentieths. That’s 347.972, and we’re still working, of course, in square centimeters.
We know that the area of the segment is the area of the sector minus the area of the triangle. So, that’s 360 minus 347.972. That’s 12.0275. Correct to two decimal places, that rounds to 12.03. And we see then that the area of the segment of our circle is 12.03 square centimeters.