A Construction of π and the Area and Circumference of a Circle
by Mike Martin

This page allows you to approximate both the area and circumference of a circle of radius r using N "triangles." The more triangles you use then the better the approximation and this page allows you to do that. To approximate the circumference of a circle we add the lengths of the blue portion of each triangle. To approximate the area of the circle we add the areas of all of the inscribed triangles. And to approximate π we note that it is defined to be the ratio of the circumference to the diameter of the circle.

This page will generate graphs associated to this approach and produce a table of values for the number of triangles, the radius, the diameter, the approximation of the area, the approximation of the circumference, and the approximation for π.   Enter the value of the parameters below, then press Evaluate.

r =

N =

The graphs are given by:

Numerical values for the exact number of triangles, radius, and diameter; also, approximations for the area, circumference, and value of π:

Let An be the area of a polygon with n equal sides inscribed in a circle of radius r.   By dividing the polygon into n congruent triangles with central angle 2π / n show that

Next, show that