How to Calculate the Surface Area of a Circle

Math Goodies

A circle is the set of all points in a plane that are equidistant from a point. This point is known as the centre of the circle, and the radius of a circle is a line segment with the circle's centre 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.

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.

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 means that 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).