𝑋𝑌𝑍 is an equilateral triangle with a side length of 92 centimeters. Three circular sectors are drawn in the triangle such that their centers are the vertices 𝑋, 𝑌, and 𝑍. The radius of each sector is 46 centimeters. Find the area of the part of the triangle bounded by the arcs of the circular sectors giving the answer to one decimal place.
Let’s begin by sketching this out. Notice how the radius of each circle is 46 centimeters. That means that each sector must reach exactly halfway along the side of the triangle. The diagram will look something like this. Ad we’re trying to find the area of the shaded part. So what do we need to work out?
Well, if we knew the area of the triangle and the area of the three individual sectors, we could find the difference between these. And it would tell us the size of the shaded area. So we recall that the area of a triangle could be found by using the formula, a half 𝑎𝑏 sin 𝑐. And the area of a sector is a half 𝑟 squared 𝜃 for a circle with a radius 𝑟 and an angle of 𝜃 in radians.
Remember, the interior angles of an equilateral triangle are each 60 degrees. And 60 degrees is the equivalent to a third 𝜋 radians. So we can say that the area of the triangle is a half multiplied by 92 squared multiplied by sin of 𝜋 over three. This is equal to 2116 root three.
And then, we have three identical sectors. So we can find the area of them by multiplying the area of one of the sectors by three. It’s three lots of a half multiplied by 46 squared multiplied by 𝜋 over three. And this is equal to 1058𝜋. The shaded area is the difference between these two values. It’s 2116 root three minus 1058𝜋, which is equal to 341.2144 and so on.
We need to give our answer to one decimal place. The first digit after decimal point is two. And the deciding digit is a one. Remember, if the deciding digit is less than five, we round our number down. One is indeed less than five.
So the shaded area correct to one decimal place is 341.2 centimeters squared.