A circle is a round plane figure with a boundary that consists of a set of points that are equidistant from a fixed point. This point is known as the center of the circle. There are several measurements associated with the circle. The *circumference* of a circle is essentially the measurement all the way around the figure. It is the enclosing boundary, or the edge. The *radius* of a circle is a straight line segment from the circle's center point to the outer edge. This can be measured from the center point to any point on the circle edge as its end points. The *diameter* of a circle is the measurement from one edge of the circle to the other, crossing through the center.

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.

Learn the value of Pi. Pi is defined as the ratio of a circle's circumference to its diameter. It is a constant ratio of the circumference to the 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. However, the decimal places of Pi go on and on without a specific pattern or end, so it is best to use the value of Pi as 3.14, especially when calculating with pencil and paper.

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.

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.

Convert the equation in step 2 to calculate the area of a circle from its diameter. Since 2r = d is an unequal equation, both sides of the equal sign must be balanced. If you divide each side by 2, the result will be r = d/2, we have A = Pi(r^2) = Pi((d/2)^2) = Pi(d^2)/4.

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).