Semicircle: Definition, Perimeter & Area Formulas

Definitions

A semicircle is simply half of a circle. That means if you take a circle and slice it down its diameter, or the line that runs through the circle’s interior and includes its midpoint, you’ll end up with two semicircles. It will also be helpful later to know that a circle’s radius is simply half its diameter.

So, it’s a line segment that runs from a point on the circle’s exterior to its center. Therefore, the radius of a circle (or semicircle) is always half its diameter, and the diameter of a circle (or semicircle) is always two times the radius. This will come in handy when we’re working with semicircle formulas!

Area of a Semicircle

The area of a shape is the amount of space that it encloses. In the case of a circle, the formula for area, A, is A = pi * r^2, where r is the circle’s radius. Since we know that a semicircle is half of a circle, we can simply divide that equation by two to calculate the area of a semicircle.

So, the formula for the area of a semicircle is A = pi * r^2/2. Let’s use that formula to calculate the area of a semicircle with a radius of 8 inches. We’ll use 3.14 as an approximation of pi. So, now we plug the values into the equation.

A = 3.14 * 8^2/2. 8^2 is 64, so A = 3.14 * 64/2. Simplified, that’s A = 200.96/2, which is 100.48. Therefore, the area of this semicircle is 100.48 square inches. In other words, the space that the semicircle encloses consists of 100.48 square inches.

Okay, what about a semicircle with a 10-foot diameter? We have a little snag here. We need the radius of our area formula, but this diagram gives us the diameter. Remember from earlier, though, that the radius of a circle (or semicircle) is half its diameter. So in this case, the radius is 10/2, or 5 feet long. Now, we can plug values into our formula.

A = 3.14 * 5^2/2. 5^2 is 25, so A = 3.14 * 25/2. Simplified, that’s 78.50/2, or 39.25. The area of this semicircle is 39.25 square feet. So, we can calculate the area of any semicircle if we’re given the diameter or radius.

Perimeter of a Semicircle

The perimeter of any shape is the total length of all its sides. Since a circle has no sides, its perimeter (also known as its circumference) is the length of the circle’s exterior. The formula for a circle’s circumference, C, is C = 2 * pi * r, where r is, again, the circle’s radius.

A semicircle is half a circle, however, so we’ll need to divide the formula for perimeter by two just like we did the formula for area. The perimeter P = pi * r. But wait! If we divide the perimeter formula by two, we’re only measuring the curvy part of the semicircle. We’re missing its bottom!

Since a semicircle, unlike a circle, does have a straight side, we need to account for that if we’re going to find its total perimeter, P; that side is actually its diameter, so we just need to add the diameter to the formula we’ve already determined, so that P = pi * r + d.