Here’s the excellent Steven Strogratz giving an elegant explanation for why the formula for the area of a circle works and explaining the mysteries of calculus at the same time:

The area of a circle (the amount of space inside it) is given by the formula

A = πr².

Here A is the area, π is the Greek letter pi, and r is the radius of the circle, defined as half the diameter.

All of us memorized this formula in high school, but where does it come from?

It’s not usually proven in geometry class. If you went on to take calculus, you probably saw a proof of it there, but is it really necessary to use calculus to obtain something so basic? Yes, it is.

What makes the problem difficult is that circles are round. If they were made of straight lines, there’d be no issue. Finding the areas of triangles and squares is easy. But working with curved shapes like circles is hard. The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works . . . as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small. That’s the crucial idea behind all of calculus.

Here’s one way to use it to find the area of a circle. Begin by chopping the area into four equal quarters, and rearrange them like so.

The strange scalloped shape on the bottom has the same area as the circle, though that might seem pretty uninformative since we don’t know its area either. But at least we know two important facts about it.

First, the two arcs along its bottom have a combined length equal to half the circumference of the original circle (because the other half of the circumference is accounted for by the two arcs on top). Since the whole circumference is pi times the diameter, half of it is pi times half the diameter or, equivalently, pi times the radius r.

That’s why the diagram above shows πr as the combined length of the arcs at the bottom of the scalloped shape.

Second, the straight sides of the slices have a length of r, since each of them was originally a radius of the circle. Next, repeat the process, but this time with eight slices, stacked alternately as before.

The scalloped shape looks a bit less bizarre now. The arcs on the top and the bottom are still there, but they’re not as pronounced. Another improvement is the left and right sides of the scalloped shape don’t tilt as much as they used to.

Despite these changes, the two facts above continue to hold: the arcs on the bottom still have a net length of πr and each side still has a length of r. And of course the scalloped shape still has the same area as before — the area of the circle we’re seeking — since it’s just a rearrangement of the circle’s eight slices.

As we take more and more slices, something marvelous happens: the scalloped shape approaches a rectangle. The arcs become flatter and the sides become almost vertical.

In the limit of infinitely many slices, the shape is a rectangle.

Just as before, the two facts still hold, which means this rectangle has a bottom of width nr and a side of height r. But now the problem is easy.

The area of a rectangle equals its width times its height, so multiplying πr times r yields an area of πr² for the rectangle.

And since the rearranged shape always has the same area as the circle, that’s the answer for the circle too!

What’s so charming about this calculation is the way infinity comes to the rescue. At every finite stage, the scalloped shape looks weird and unpromising. But when you take it to the limit — when you finally get to the wall — it becomes simple and beautiful, and everything becomes clear.

That’s how calculus works at its best.

Steven Strogratz –

The Joy of X(recommended!)