The three primary characteristics of a circle are its circumference, diameter and radius. All circles share common properties that allow for formulas that relate these characteristics to one another. For example, the famous number pi (approximately 3.14, or a bit more precisely, 3.14156) is the ratio of a circle's circumference to its diameter, and this ratio holds true for all circles. It is also true that a circle's circumference has a specific relationship with its radius, and this means that there is a simple formula for calculating the radius of a circle if you know its circumference.
The circumference of a circle is the distance around a circle's edge. It is what you draw if you use a standard pin-and-pencil compass to draw a circle around a central point. The circumference of any circle is directly proportional to the diameter and the radius of the circle.
The radius of a circle is a line drawn from the direct center of the circle to its outer edge. A radius can be drawn in any direction from the central point. A circle's radius is exactly half the length of the same circle's diameter, which is a line that divides the circle into two equal halves.
The Relationship of Circumference and Radius
The definition of pi reveals the equation for the circumference of a circle. Pi is equal to the circumference of a circle divided by its diameter. In mathematical terms this looks like the following:
pi = C / d
You get the equation for circumference by solving for C in the equation above.
C = pi x d
And since the diameter of a circle is twice as long as its radius, you can substitute 2r for d, with r standing for radius.
C = pi x 2r
Calculating Radius Using Circumference
If you know the circumference of a circle, you can use the equation for circumference to solve for the radius of that circle. First you have to rearrange the equation to solve for r. Do this by dividing both sides by pi x 2. This operation will cancel on the right side of the equation and leave r by itself. If you then flip the sides of the equation, it will look like this:
r = C / (pi x 2)
Suppose you know that the circumference of a circle is 20 centimeters and you want to calculate the radius. Just plug the value for the circumference into the equation and solve. Remember that pi is approximately equal to 3.14.
r = 20 cm / (3.14 x 2) = 3.18 cm