Area of Semicircle

A semicircle is half of a circle - a shape like this:

It is formed by drawing the diameter of a circle - a straight line the runs from one point on the circle's circumference, through the center of the circle, and to another point on the circle's circumference on the other side.

The diameter splits the circle into two equal semicircles.

As we have discussed before, a circle is a geometric shape that is completely defined (in terms of its area or perimeter) by one property- its radius. The area of a circle is given by the formula A_circle=π*r², where π  is a special number, which is the same for all circles, that is the ratio between a circle’s diameter and its circumference.

Since a semicircle is half of a circle- its area is simply half the area of a full circle: A_semicircle=π*r²/2.

Let's look at a simple geometry problem involving semicircles and their area.

Problem

A semicircle has a perimeter whose length is 8 units. Find its area.

Strategy

The formulas for the area and circumference of circles (and thus semicircles) are very simple. They only require knowing the length of the radius. So we will solve this using the simple formula for the area of a semicircle (A_semicircle=π*r²/2). We will get the length of the radius from the perimeter.

The perimeter of a full circle is C_circle=2*π*r. So half a circle would be just π*r - but we also have to add the straight part forming the semicircle to get the complete perimeter. That part is simply the diameter, which is twice the radius. So P_semicircle=π*r+2r=(2+π)r. Here, P_semicircle=8, so (2+π)r=8 and r=8/(2+π).

Now that we have r, we plug it into the Area formula, and have the answer:

A_semicircle=π*r²/2 = π*[8/(2+π)]² = π*[64/(4+4π+π²)]