Question

Why is the area of a circle `pi r^2`?

Answer 1

See explanation...

Explanation:

The circumference of a circle of radius `r` is of length `2pi r`, by the definition of `pi`.

If you dissect a circular disc into segments of equal size, you can reassemble them head to tail into a sort of bumpy parallelogram with height approximately `r` and base width approximately `pi r`, being half of the circumference of the circle. Hence the area is `pi r^2`.

It works better the smaller your segments are, but here's an animation I made for `8` segments: