Chapter 2 | Limits
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Deriving the Formula for the Area of a Circle 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 triangles goes to zero, you can obtain the area of the circle. To see this, carry out the following steps: 1. Express the height h and the base b of the isosceles triangle in Figure 2.31 in terms of θ and r .
Figure 2.31 2. Using the expressions that you obtained in step 1, express the area of the isosceles triangle in terms of θ and r . (Substitute (1/2)sin θ for sin( θ /2)cos( θ /2) in your expression.) 3. If an n -sided regular polygon is inscribed in a circle of radius r , find a relationship between θ and n . Solve this for n . Keep in mind there are 2 π radians in a circle. (Use radians, not degrees.) 4. Find an expression for the area of the n -sided polygon in terms of r and θ . 5. To find a formula for the area of the circle, find the limit of the expression in step 4 as θ goes to zero. ( Hint: lim θ →0 (sin θ ) θ =1). The technique of estimating areas of regions by using polygons is revisited in Introduction to Integration .
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