A circle is the set of all points in a plane that are equidistant from a point. This point is known as the center of the circle, and the radius of a circle is a line segment with the circle's center and any point on the circle as its end points. The surface area of a two-dimensional closed curve such as a circle is the total area contained by that curve. The area of a circle may be calculated when the length of its radius, diameter or circumference is known.

## Figuring the Area

### Step 1

Learn the value of Pi. Pi is defined as the ratio of a circle's circumference to its diameter. This means that Pi = c/d where c is the circumference of a circle and d is its diameter. The exact value of Pi can never be known, but it can be estimated to any desired accuracy. The value of Pi to six decimal places is 3.141593.

### Step 2

Examine the formula for the area of a circle. It's A = Pi(r^2) where A is the area of the circle and r is the radius of the circle. Archimedes proved this in approximately 260 B.C. using the law of contradiction, and modern mathematics does so more rigorously with integral calculus.

### Step 3

Use the equation obtained in step 2 to calculate the area of a circle with a known radius. A circle with a radius of 2 has an area of A = Pi(r^2) = Pi(2^2) = 4 x Pi, or approximately 12.57.

### Step 4

Convert the equation in step 2 to calculate the area of a circle from its diameter. Since 2r = d means that r = d/2, we have A = Pi(r^2) = Pi((d/2)^2) = Pi(d^2)/4.

### Step 5

Convert the equation in step 4 to calculate the area of a circle from its circumference. We know that Pi = c/d from step 1 so d = c/Pi. Substituting this value for d into A = Pi(d^2)/4, we have A = Pi((c/Pi)^2)/4 = c^2/(4 x Pi).