The length of a chord is 43 centimeters and the radius of the circle is 26 centimeters. Find the area of the circular segment giving the answer to the nearest square centimeter.
Let’s begin by sketching a diagram of this scenario out. We have a circle with a chord of length 43 centimeters. Let’s add the center of the circle, 𝑜, and then we know the line joining 𝑜 to the end of the chords must be the radius. So that’s 26 centimeters. The question wants us to find the area of the circular segment, giving our answer to the nearest square centimeter. The segment is this piece that looks a little bit like this segment of an orange. I’ve shaded it in yellow. And so, let’s recall how we find the area of a segment.
We begin by finding the area of the sector. Now, for a sector with a radius 𝑟 and an angle 𝜃, that’s 𝜃 over 360 times 𝜋𝑟 squared. We then subtract the area of the triangle. Now we’re going to use the trigonometric formula a half 𝑎𝑏 sin 𝑐. We replace 𝑎 and 𝑏 with 𝑟 and 𝑐 with 𝜃. And so, the area of this triangle is a half 𝑟 squared sin 𝜃. And so, we see that the area of a circular segment is 𝜃 over 360 times 𝜋𝑟 squared minus a half 𝑟 squared sin 𝜃.
Now, we do indeed know the radius of our circle; it’s 26 centimeters. What we don’t yet know, however, is the size of the angle of our sector. We’ll call that 𝜃. And how we’re going to calculate it? Well, we see we’re looking to find the angle. And we know the length of all three sides. But it isn’t a right-angled triangle. So we’re going to use non-right-angle trigonometry. In fact, we’re going to use the cosine rule. This says that 𝑎 squared equals 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝑎. Its rearranged form, to help us find an angle, is cos 𝑎 equals 𝑏 squared plus 𝑐 squared minus 𝑎 squared over two 𝑏𝑐.
Substituting what we know about our triangle into this formula, and we get cos 𝜃 equals 26 squared plus 26 squared minus 43 squared over two times 26 times 26. Well, the fraction on the right becomes negative 497 over 1352. Now, this is, of course, equal to cos 𝜃. We take the inverse cos of both sides. And we find 𝜃 is equal to 111.56 degrees. Now, we’re going to try to use the unrounded version of this for accuracy.
Now that we know the size of angle 𝜃 and the radius of our circle, we can substitute this all into our formula. The area of the sector is 111.56 over 360 times 𝜋 times 26 squared. And the area of the triangle is a half times 26 squared times sin 111.56. That gives us 343.828 and so on. The question tells us to give then our answer correct to the nearest square centimeter. And so, we see that the area of the circular segment — and in this case, in fact, it’s the minor circular segment — is 344 centimeters squared.