Circle Area

Area enclosed by a circle of radius $R$ is $S = \pi \cdot R^2$. Let's make sure of this, using the ability to calculate the area of a rectangle.

Divide the circle in two halves with its diameter. Divide each half into equal sectors. ”Opening” the halves and inserting one into the other, one gets a shape of the same area that of the initial circle. This shape is almost a rectangle. Almost — because the longer sides are not quite straight. These sides are half the length of the circle long, that is, $\pi \cdot R$. The short side of the shape obtained is exactly as long as the radius of the initial circle. Rectangle area is given by the product of its sides' lengths: $S≈(\pi \cdot R)\cdot R = \pi \cdot R^2$.

We've used the formula for the area of a rectangle, though the shape obtained is not exactly a rectangle, thus the approximate equality sign. It is clear though that if the circle is divided into a larger number of equal parts, the difference from a rectangle would get lesser and lesser. In the limit, the shape wouldn't differ from a rectangle, so this model is not only illustrative, but also quite valid.

The model can be manufactured from wood and a stripe of leather. The latter should be picked of a different color than the wood's, so that the circumference and the rectangle's long sides were clearly distinguishable. In one of the two parts, one sector should be divided in two halves — so that the external pieces were halves of standard sectors. Then the shape obtained would look more like a rectangle when put together, otherwise — like a parallelogram.