Area of a circle. Reducing to the area of a triangle

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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.

Assemble a circle of concentrically arranged stripes of, say, leather. The external one should be the longest, next one — a bit shorter etc. Lengths should be chosen so that the circle is formed when wrapping a circle. The stripes' ends meet along one radius.

Now unwrap all the stripes at once and the circle turns into almost a triangle. "Almost" — because its legs are not straight lines, but are comprised of steps. Several formulae for finding the area of a triangle are studied at school (and all of them provide the same answer!). Let us use one of those — the area of a triangle equals half of a product of side length (of the base, for example) by the length of its altitude. The base length in our case exactly equals the length of the initial circle's circumference, that is $2 \cdot \pi \cdot R$. The altitude is simply the circle's radius. Thus the area of the shape is $S ≈ (1/2) \cdot (2 \cdot \pi \cdot R) \cdot R ≈ \pi \cdot R^2$.

We've used a formula for the area of triangle, though the shape obtained is not exactly a triangle, that's why the equality is approximate. It is nevertheless clear that if the circle is composed of thinner and thinner stripes, the steps on triangle's legs will be smaller. And in the limit the shape will not differ from a triangle, so the reasoning is quite valid.