If we cut the circle into quarters, we can approximate the area of each quarter by the areas of two triangles: the one with the red dotted line as its base, and the one with the blue dotted line as its base. The altitude or height of the latter is of course, r, the radius of the circle. We can even use trigonometry to calculate the height of the first one, it is: r sin 45° = r √2 / 2.
These don't seem to be particularly good estimates, but notice that the area we want must lie somewhere between the areas of the two circles. The blue base is larger, and the red base is smaller. And something else: we don't have to use quarters, instead we can cut the circle up into a large number of thin triangles each with sides of length r.
As we make n, the number of triangles larger and larger, then in the limit as n gets very large, the altitude of each triangle approaches r, the radius, and the base of the triangle approaches the length of the arc of the circle's circumference lying between the vertices of the triangle. Let's call that length b for base. Then the area of the small triangle is
1/2 * r * b
And the area of the whole circle's worth of small triangles is
A = 1/2 * r * n * b
But n * b is just the circumference of the circle = 2 * pi * r. Substituting
A = 1/2 * r * 2 * pi * r = pi * r2
Now, don't get me wrong, I have nothing against Euclid. But I would bet this proof is way older than him. It's a kind of corollary of Stigler's Law.
