A circle has a radius of 10 centimeters, and the central angle of a circular segment is 25𝜋 over 28 radians. Find the area of the circular segment giving the answer to two decimal places.
In this question, we’ve been given information about the radius of a circle and the central angle of its circular segment. If we sketched this out, it looks a little something like shown. But of course, the shape we’ve sketched is the sector of the circle. We’re looking to find the area of the circular segment. That’s the bit that looks a little bit like the slice of an orange, and it’s shaded as shown.
So, how do we find its area? Well, in fact, we think of it as a little bit like a composite shape. To calculate its area, we begin by calculating the area of this sector. We then subtract the area of this triangle, and that will leave simply the area of the segment we require. So, we begin by recalling the area of a sector.
For a circle with a radius 𝑟 on a sector with an angle of 𝜃 degrees, its area is 𝜃 over 360 times 𝜋 squared. We’re working in radians though, so we can use the slightly nicer formula a half 𝑟 squared 𝜃. Once again, 𝑟 is the radius. But this time, this angle, 𝜃, is measured in radians. We’ll use the formula to help us find the area of a nonright-angled triangle. It’s a half 𝑎𝑏 sin 𝑐. So, we begin by finding the area of the whole sector. It’s a half times the radius squared. That’s 10 squared times 25𝜋 over 28. That gives us 625𝜋 over 14.
Then, if we label our triangle so that the angle is at vertex 𝑐, the area of the triangle is then a half times 10 times 10 or 10 squared times sin of 25𝜋 over 28. This time, we get 16.513 and so on. We said that the area of the segment was the area of the sector minus the area of the triangle. So, that’s 625𝜋 over 14 minus 16.513. And, if possible, we use this exact value.
Typing this into our calculator and we get 123.7357 and so on. But the question asks us to round our answer to two decimal places. That’s 123.74. And so, we see that the area of our circular segment is 123.74 square centimeters.