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.