Math textbooks often present formulas, even extremely important ones, without proof or justification. That’s probably fine for the student with average interest and ability, but it is often a disservice to students who really want to understand the subject. My preference would be that the books have derivations in a sidebar, with the understanding that students will not be tested on that material, but it will be there for those students who want to understand the material on a deeper level.

An example of such a formula is the one for the area of a circle: . The formula was derived by Archimedes over 2,000 years ago; it is certainly one of the most useful results in mathematics. In spite of its fundamental importance, several high school math books that I have checked didn’t attempt to show where it comes from. In this post I will explain one way that the formula can be derived, and in other posts I will explain a number of other important formulas.

First, imagine that we divide the circle into equal-sized triangles as show below.

We can approximate the circle’s area by just computing the area of all the triangles. If there are n triangles, then \(\mbox{circle area }\approx n\cdot\frac{1}{2}bh \)

Now imagine we begin increasing the number of triangles. Two things will happen:

• The area approximation will become closer and closer to the real value
• The altitude h of each triangle will become arbitrarily close to R, the circles radius.

So
\[\mbox{circle area}\approx\:n\:\cdot\:\frac{1}{2}\,bR,\:or\:\frac{1}{2}\,R\:\cdot\:nb. \]

Notice the product nb – for very large n, this become the same as the circumference of the circle, or \(2\pi R\)

If we replace nb in the above formula, we finally have,
\[\mbox{circle area}\approx\:\frac{1}{2}\,R\,\cdot\,2\pi\,R\:=\:\pi\,R^{2} \]

So, as n becomes infinite, the area actually becomes \(\pi R^{2} \)

Now, for those who have been exposed to only algebra and geometry, the type of reasoning used here may seem strange. We are taking an infinite number of triangles, each infinitely small, and coming up with a finite number – the exact area of the circle.

This is basically what the subject of integral calculus is about. Integral calculus provides very general and powerful methods for computing areas. And it’s not just about finding areas of geometric shapes – an amazing number of useful problems, if thought of the right way, are equivalent to the problem of finding an area.

Calculus was not invented until the late 1600’s, so Archimedes couldn’t use it in deriving the circle area formula. The method he actually used was roughly equivalent to the derivation used here.

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This entry was posted on Monday, June 29th, 2009 at 12:51 am and is filed under Mathematics History, Thrilling Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.