The chord and the radius of a circle both measure 33 centimeters. Find the area of the major circular segment, giving the answer to two decimal places.
We’ve been given some information about the length of the chord and the radius of a circle. So let’s sketch this out. Let’s imagine the cord is given by the line segment 𝐴𝐵 and the center of our circle is 𝐶. By definition, 𝐴𝐶 and 𝐵𝐶 are the radii of our circle. So all three sides of the triangle we’ve drawn are 33 centimeters. Now, the question wants us to find the area of the major circular segment. But which bit of the circle is this? Well, the segment of a circle usually looks like a slice of orange. The minor segment of this circle is the piece shaded in pink. And that means then that the major circular segment is everything else. This is what we’re looking to find the area of.
Notice that, in fact, we have a composite shape. We have a triangle, and, in fact, that’s an equilateral triangle. And then we have a sector. So let’s begin by finding the area of our triangle. We recall that we have a trigonometric formula for the area of a triangle 𝐴𝐵𝐶. It’s a half 𝑎𝑏 sin 𝑐. Well, we can see that both 𝑎 and 𝑏 are the radii of our circle; they’re 33 centimeters. But what’s 𝑐? Well, our triangle is an equilateral triangle, and we know that one angle in an equilateral triangle is 60 degrees or 𝜋 by three radians. Let’s work in radian measure. So we see that the area of the triangle is a half times 33 times 33 times sin of 𝜋 by three. That’s a half times 1089 times the square root of three over two, which is 1089 root three over four.
Now, the question wants us to give our answer to two decimal places, but we’re not quite finished. So for now, we’re going to leave this in exact form. And so we say that the area of our triangle is 1089 root three over four square centimeters. Our next job is to find the area of the sector. Now, let’s imagine we’ve been given a sector with an angle of 𝜃 and a radius 𝑟. If we’re working in degrees, the area is 𝜃 over 360 times 𝜋𝑟 squared. Essentially, we’re working out a proportion of the area of the whole circle. If we’re working in radian measure, though, there is a slightly less complicated formula. The area is a half 𝑟 squared 𝜃. And it’s important to realize that 𝜃 can be an acute, obtuse, or a reflex angle.
In this case, 𝜃 is the size of the reflex angle 𝐴𝐶𝐵. Since we’re choosing to work in radians and we know angles about a point add up to two 𝜋 radians, we can say that 𝜃 is equal to two 𝜋 minus 𝜋 by three. We write two 𝜋 as six 𝜋 by three, and we see that 𝜃 is equal to five 𝜋 by three radians. Now, we know that the radius of our circle is 33 centimeters and the angle 𝜃 is five 𝜋 by three radians. So the area of our sector, and that’s the bit shaded in pink, is a half times 33 squared times five 𝜋 by three. That’s 1815𝜋 over two. And so the exact area of our sector is 1815𝜋 over two square centimeters.
The area of the major circular segment is the sum of these two values. It’s 1089 root three over four plus 1815𝜋 over two. That’s 3322.54 and so on. We’re looking to give our answer to two decimal places, so that’s 3322.55. And that’s square centimeters.