The figure shows square 𝐴𝐵𝐶𝐷 inscribed in a circle 𝑀. Given that the radius of circle 𝑀 is 32 centimeters, find the perimeter of the shaded region. Give your answer to the nearest centimeter.
So what I’ve done first of all is marked on diagram that we’ve got 32 centimeters as the radius. So therefore, what we can do is work out what the line 𝐷𝐵 is going to be, so the diagonal of our square. This is gonna be equal to two multiplied by 𝑀𝐵 because 𝑀𝐵 is the radius, and 𝐷𝐵 represents a diameter. Well, this is gonna give us two multiplied by 32 which is equal to 64 centimeters. Okay, great. So this is the diagonal of our square. But is this useful? Well, yes, it is because what we need to do is find the perimeter of the shaded region. And to do that, it’s gonna be two sections, first of all, the arc from 𝐶 to 𝐵. And then we have the line which is one of the sides of our square.
But how can we work out a side of the square? So actually, to find out what 𝑆 is or one of the side lengths, we can use the Pythagorean theorem. And this is because if we take half of our square, we’ve got a right triangle with two side lengths 𝑆 and a hypotenuse we know of 64. So therefore, if we substitute what we’ve got into our Pythagorean theorem, we’ve got 64 squared is equal to 𝑆 squared plus 𝑆 squared, which is gonna give us 4096 is equal to two 𝑆 squared. So if we divide this by two to find out what 𝑆 squared is, we’re gonna have 2048 is equal to 𝑆 squared. And then what we’re gonna do is take the square root of both sides of the equation. So we’re gonna have that the distance 𝑆, so the side length of our square, is equal to the square root of 2048.
At this point, I’m gonna leave it like this. And that’s because we want to keep it as accurate as possible so we don’t have any rounding errors. So now, what we need to do is find the arc length. Well, to find the arc length, what we need to notice is the fact that our circle is broken up by four identical sides because we have a square. So therefore, the arc length 𝐶𝐵 is gonna be the same as the arc lengths 𝐶𝐷, 𝐷𝐴, and 𝐴𝐵. So therefore, to find out what our arc length is going to be, what we need to do is have the circumference of a circle, which is 𝜋d or two 𝜋𝑟, and then divide it by four because we have four identical arcs.
So we can say that the arc length 𝐶𝐵 is equal to 𝜋 multiplied by 64, because that’s our diameter, divided by four, which is gonna give us an answer of 16𝜋. Okay, great. So we now had the two sections that we need to put together to find the perimeter of the shaded region. So let’s do that now.
So if we’ve got the perimeter of the shaded region, this is gonna be equal to the square root of 2048 plus 16𝜋, which is gonna give 95.5203. But we’re looking for the answer to the nearest centimeter. So therefore, we can say that the perimeter of the shaded region is gonna be equal to 96 centimeters. And that’s to the nearest centimeter.