Circumference of a Circle

In traditional approaches to mathematics, the circumference of a circle has not always been clearly defined. Sometimes the circle itself was called the circumference, and at other times, the measure of the distance around the circle was called the circumference. If we define the circumference as the perimeter of the circle the measure of the circle is symbolized by the formula c= πd or c= 2πr.

There exist more precise definitions of a circumference. To arrive at a more precise definition it is necessary to introduce the concept of limits. By using the limit concept, the circumference of a circle may be defined as the limit of the perimeter of an inscribed regular polygon. To illustrate this we can first inscribe a square in a circle. The sum of the sides of the square will be an approximation of the circumference of the circle. Then, bisecting the central angles, which are subtended by the sides of the square we can inscribe a regular octagon. The sum of the sides of the octagon will be a closer approximation of the circumference. Next bisecting the central angles subtended by the sides of the octagon, we can inscribe a regular 16-gon. The sum of the sides of the 16-gon will be an even closer approximation of the circumference. By a similar process we can then inscribe a regular 32-gon and 64-gon and so on. Clearly the sum of n sides of an inscribed regular n-gon can be made to approximate the circumference of the circle as closely as desired by choosing n sufficiently large.

In other words the circumference of a circle may be defined as the limit of the perimeter of an inscribed regular n-gon as n increases.

UNIT 6

“It appears to me that if one wants to make

progress in mathematics, one should study

the masters and not the pupils.”

N.H. Abel