Area of a circle (Part 3)

Calculus, Geometry, Theorems 1 Minute

Math majors are completely comfortable with the formula A = \pi r^2 for the area of a circle. However, they often tell me that they don’t remember a proof or justification for why this formula is true. And they certainly don’t remember a justification that would be appropriate for showing geometry students.

In this series of posts, I’ll discuss several ways that the area of a circle can be found using calculus. I’ll also discuss a straightforward classroom activity by which students can discover for themselves why A = \pi r^2.green lineIf R denotes a circular region with radius a centered at the origin, then

A = \displaystyle \iint_R 1 \, dx \, dy

This double integral may be computed by converting to polar coordinates. The distance from the origin varies from r=0 to r=a, while the angle varies from \theta = 0 to \theta = 2\pi. Using the conversion dx \, dy = r \, dr \, d\theta, we see that

A = \displaystyle \int_0^{2 \pi} \int_0^a r \, dr \, d \theta

A = \displaystyle \int_0^{2\pi} \left[ \frac{r^2}{2} \right]_0^a \, d\theta

A = \displaystyle \int_0^{2\pi} \frac{a^2}{2} \, d\theta

A = \displaystyle 2 \pi \cdot \frac{a^2}{2}

A = \displaystyle \pi a^2

We note that the above proof uses the fact that calculus with trigonometric functions must be done with radians and not degrees. In other words, we had to change the range of integration to [0,2\pi] and not [0^o, 360^o].

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Published by John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

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One thought on “Area of a circle (Part 3)”

  1. Pingback: Area of a Circle: Index | Mean Green Math

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