Video Transcript
A chord of length 90 centimeters is 42 centimeters away from the center of its circle. Find the area of the minor circular segment, giving the answer to the nearest square centimeter.
We’ve been given some information about the chord of a circle. Let’s begin by sketching a diagram of this circle out. We have a chord; let’s call that 𝐴𝐵. We know that its length is 90 centimeters. Now, we’re also told it’s 42 centimeters away from the center of the circle; let’s call the center 𝑂. And then, we know that the shortest distance from 𝑂 to the chord 𝐴𝐵 must be 42 centimeters.
Well, we know that the shortest distance from a line to a point is found by constructing the perpendicular bisector of that line. And so, we know that this angle here must be a right angle. Now, we’re looking to find the area of the minor circular segment; that’s the area shaded in pink. And so, to do so, we’re going to need to calculate both the length of the radius of our circle and the angle of the sector. Now, I’ve called that 𝜃. Let’s draw one of the right-angled triangles from our diagram a little bit bigger so we can see what we need to do next.
We know that one side of the triangle is 42 centimeters and the other side is half of 90; it’s 45 centimeters. Now, this length here will be 𝑟 or 𝑟 centimeters, where 𝑟 is the radius. We can also see that this angle must be half 𝜃. Let’s begin by finding the length of the radius. Notice we have a right-angled triangle. So, we can use the Pythagorean theorem to find the length of 𝑟. This says that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, if we have a triangle whose hypotenuse is 𝑐 units and the other two sides are 𝑎 and 𝑏 units in length, we say that 𝑎 squared plus 𝑏 squared equals 𝑐 squared. Now, in this case, our hypotenuse is 𝑟 centimeters. So, we can say that 42 squared plus 45 squared must be equal to 𝑟 squared. 42 squared plus 45 squared is 3789. 𝑟 is equal to the square root of this. That’s three times the square root of 421.
And so, we now know the length of the radius of our circle, but what about 𝜃? Well, this time, we’re going to use right angle trigonometry. If we label a half 𝜃 as our included angle, then the side opposite to this is 45 centimeters and the side adjacent is 42 centimeters. The tan ratio says that tan 𝜃 is equal to opposite over adjacent. Well, we defined 𝜃 to be equal to the angle of our sector. So, in this case, we say that tan of a half 𝜃 is 45 over 42.
To solve for 𝜃, we take the inverse tan of both sides and then multiply through by two. So, we find 𝜃 is equal to two times the inverse tan of 45 over 42, which is equal to 93.94 degrees. Now, notice, we’re not going to round; we’re actually going to use this exact value for 𝜃 in our later calculations. So, now, we know the radius of our circle and the angle of the sector, how do we find the area of the segment?
Well, if we look carefully, we see that the area of our segment is found by finding the area of the sector and then taking away the area of the triangle. Well, the area of a sector of angle 𝜃 is 𝜃 over 360 times 𝜋𝑟 squared. And we can use the formula a half 𝑎𝑏 sin 𝑐 to find the area of the triangle. That’s a half 𝑟 squared sin 𝜃. Let’s substitute everything we know about our sector into the formula.
Notice how we’re using the value of 𝑟 squared rather than the value of 𝑟. And we’re going to carry forward the value of 𝜃 that we calculated. So, the area of the segment is 93.94 and so on out of 360 times 𝜋 times 3789 minus a half times 3789 times sin of 93.94. That gives us 1216.47 and so on. Now, we’re told to give our answer to the nearest square centimeter. So, this rounds to 1216. And we can, therefore, say that the area of the minor circular segment is 1216 square centimeters.
Now, it’s important to realize that we were asked to find the area of the minor circular segment, in other words, the smaller segment in the circle. Had we been asked to find the major circular segment, this would’ve been the largest segment in our circle.
