Video Transcript
A circular flower bed is divided into four parts by an equilateral triangle inscribed in the circle. The radius of the flower bed is nine meters. Find the area of each minor circular segment, giving the answer to two decimal places.
Let’s begin by sketching this flower bed. It is a circle, and then there is an equilateral triangle inscribed within it, which means that the three vertices of this triangle all lie on the circumference of the circle. So the flower bed looks like this. And we can see that it is indeed divided into four parts, as specified in the question. Now, each side of this triangle is a chord of the circle, and so it divides the circle into a minor segment and a major segment. And it is the area of each minor circular segment that we’re asked to find.
As the triangle is equilateral, all three chords are the same length. And so all three minor circular segments are congruent. Let’s just focus on this one shaded in orange. To help us see how we can find this area, let’s also sketch in the radii from each vertex of the triangle to the center of the circle. We can then recall that the formula for calculating the area of a segment, which has a radius of 𝑟 and a central angle of 𝜃 measured in degrees, is 𝑟 squared over two multiplied by 𝜋𝜃 over 180 minus sin of 𝜃 degrees. This is a simplification of a method in which we first calculate the area of a circular sector and then subtract the area of the triangle formed by the two radii and the chord connecting the endpoints of the radii.
In this question, we’re told that the radius of the flower bed is nine meters. So in order to apply this formula, we just need to calculate the measure of the central angle of the sector. Well, as the original triangle was equilateral, the three triangles we have divided it into are congruent as each have two sides which are the radius of the circle and one side which was the original side length of the triangle. The three central angles are therefore all the same. And as together they form 360 degrees at the center of this circle, each angle is one-third of 360 degrees. So the value of 𝜃 is 360 over three, which is 120.
We can now substitute 𝑟 equals nine and 𝜃 equals 120 into the formula for finding the area of a segment. We have nine squared over two multiplied by 120𝜋 over 180 minus sin of 120 degrees. Nine squared is 81. 120𝜋 over 180 can be simplified to two 𝜋 over three. And sin of 120 degrees is root three over two.
Now, the question specifies that we should give our answer to two decimal places. So we need to evaluate 81 over two multiplied by two 𝜋 over three minus root three over two on our calculators, which is 49.748 continuing. The digit in the third decimal place is an eight, so we’re rounding up to 49.75. The units for the length in this question, the radius, were meters, so the units for the area will be square meters. We found then that the area of each of the minor circular segments in this flower bed is 49.75 square meters correct to two decimal places.
