Question Video: Finding the Perimeter of a Semicircle given Its Area Mathematics

Video Transcript

The area of the given semicircle is 51.04 centimeters squared. Find the perimeter of the semicircle to the nearest centimeter.

We recall that the formula for finding the area of a semicircle is 𝜋𝑟 squared over two, where 𝑟 is the radius of the circle. 𝜋𝑟 squared gives the area of the full circle, and then we divide it by two. The perimeter of a semicircle is made up of a straight edge, which is the circle’s diameter, and a curved portion, which is half of the circle’s circumference. So the perimeter of the semicircle is equal to 𝑑 plus 𝐶 over two. And the formula for finding the circumference of a circle is 𝜋𝑑, 𝜋 times the diameter. The perimeter of a semicircle can be written as 𝑑 plus 𝜋𝑑 over two.

In order to determine the perimeter of this semicircle then, we first need to calculate its diameter. Using the given area of the semicircle and the formula for calculating the area of a semicircle, we can form an equation: 𝜋𝑟 squared over two equals 51.04. If we can solve this equation to determine the value of 𝑟, we can then calculate the diameter of the circle by recalling that the diameter is twice the radius. We begin by multiplying both sides of the equation by two to give 𝜋𝑟 squared equals 102.08.

Next, we divide both sides of the equation by 𝜋, giving 𝑟 squared equals 102.08 over 𝜋, and then take the square root of each side of the equation. Evaluating gives 𝑟 equals 5.7002 continuing. So we found the radius of the circle, and we can double it to find the diameter, which gives 11.4005 continuing.

We’re now able to substitute this value of 𝑑 into our formula for the perimeter of a semicircle. And we need to make sure we keep this value as exact as possible. You may want to save this value in the memory of your calculator so that you can then recall it directly when evaluating the perimeter. Substituting this exact value of 𝑑 and then evaluating gives 29.3084. We’re asked for the perimeter to the nearest centimeter, so we round down to 29. By recalling the formulae for calculating both the area and perimeter of a semicircle then, we found that the perimeter of this semicircle to the nearest centimeter is 29 centimeters.