The area of a circular sector is five-eighths of the area of the circle. The radius is 27 centimeters. Find the central angle in radians giving the answer to two decimal places and the perimeter of the sector giving the answer to the nearest centimeter.
Let’s sketch this out to get an idea what’s going on. We start with the circle. We know that the sector area is five-eighths the circle area. Five-eighths is a little bit more than half. If the sector was four-eighths, if it was one-half of the circle, it would be this much. But we have five-eighths, so we’ll make it a little larger. We can identify the sector and label the radius.
We need to know what this angle is. For now, we can call it 𝜃. And we’re interested in the angle 𝜃 given in radians. In radians, a full turn equals two 𝜋. Two 𝜋 is the radian measure of 360 degrees. To find our central angle, we know that we’re going five-eighths of the way around the circle. We’re interested in radians. And that means we want five-eighths times two 𝜋. Multiplying that together, we can reduce a little bit. Two goes into eight four times. Our central angle 𝜃, in radians, is five 𝜋 over four.
However, we need to give this answer to two decimal places. So, we’ll plug five 𝜋 divided by four into the calculator. And we’ll get 3.9269 continuing. To round this to the second decimal place, we’ll look to the deciding digit on the right. This six is larger than five. We need to round up. So, we get the central angle 𝜃 is 3.93 radians.
We’re also interested in the perimeter of this sector. The perimeter will be the distance around this sector, which is a radius plus a radius plus an arc length. We usually use the variable 𝑠 to represent the arc length. To find that arc length, we start with our central angle in radians and we multiply it by the radius. I’m gonna plug in five 𝜋 over four for our central angle. You could also use this rounded value. But since we’re already going to round at the end of this step, it’s best to use a value that hasn’t already been rounded. And the radius is 27.
So, we need to multiply five 𝜋 over four times 27. And we’ll get the arc length equals 106.028 continuing. If we know that the perimeter equals the radius plus the radius plus the arc length, we plug in 27 for the radius and 106.028 continuing for the arc length. The best way to do this is to keep this value, this answer, in your calculator and just add 27 plus 27 to it. This will allow us to round in the final step.
Now we have 160.028. Rounding to the nearest centimeter is the nearest whole number. The digit on the right, the deciding digit, is a zero. So, we round down. The perimeter equals 160. And our units were being measured in centimeters. That means the perimeter, the distance around this sector, is 160 centimeters.