Chapter 2.3, Problem 5SP

Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. The Greek mathematician Archimedes (ca. 287—212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit.

We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think of the regular polygon as being made up of n triangles. By taking the limit as the vertex angle of these mangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps:

5. To find a formula for the area of the circle, find the limit of the expression in step 4 as $\theta$ goes to zero. (Hint: $\underset{\theta \to 0}{\lim }\frac{\left(\sin \theta \right)}{\theta }=1$ .)