The diameter of a circle is 14 centimeters and the central angle is 5.79 radians. Find the area of the circular segment giving the answer to two decimal places.
Let’s begin this question by considering the circle described. We are told that the central angle is 5.79 radians. And as the diameter is 14 centimeters, the radius must be seven centimeters. The circular segment required is shown in pink. We could calculate the area of the circle and then subtract the area of the minor segment shown in orange. However, in this question, it’ll be quicker to work out the area of the major sector number one and triangle two.
The area of any sector where the angle 𝜃 is given in radians is equal to a half 𝑟 squared 𝜃. In this question, this will be equal to a half multiplied by seven squared multiplied by 5.79. This is equal to 141.855. Area one is equal to 141.855 square centimeters. The area of any triangle can be calculated using the formula a half 𝑎𝑏 multiplied by sin 𝐶.
In this question, lengths 𝑎 and 𝑏 are both seven centimeters. The angle required is the acute one inside the triangle. We know that 360 degrees is equal to two 𝜋 radians. This means that angles at a point or in a circle must add up to two 𝜋 radians. The missing angle will be equal to two 𝜋 minus 5.79. This is equal to 0.4931 and so on. Ensuring that our calculator is in radian mode, we can now calculate the area of the triangle. This is equal to 11.5991 and so on.
We can now calculate the area of the circular segment by adding our two answers. This is equal to 153.4541 and so on. As we need to give our answer to two decimal places, the four is the deciding number. This means we will round down. The area of the circular segment to two decimal places is 153.45 square centimeters.