Most people know the figuring of the area of a circle is A = pi x r squared. It was Archimedes in the 3rd century BC who did the figuring in Measurement of the Circle. But what I find important about this is not the formula but the kind of thinking that proved the point. Archimedes writes
Since then the area of the circle is neither greater than nor less that [the area of the triangle], it is equal to it.
The suppositions go: A< T or A=T or A>T. If the first and third don’t work, then the second must be true, double reductio. For kicks: 1st case
A – Area (inscribed polygon) < A - T leads to T < Area (inscribed polygon) Area (inscribed polygon) = 1/2hQ < 1/2rC = T where h = apothem, Q = the perimeter of the polygon and C = the circumference of the circle Since here T < the area of the inscribed polygon and the area of the inscribed polygon is also < T then the supposition of A < T contradicts itself
and so forth. Here I’ve paraphrased Will Dunham’s Journey through Genius: The Great Theorums of Mathematics. Archimedes proves the area and approximates pi in the same text but does so by a wonderful bit of questioning and analysis. The area is important, but without the skill of bringing it all together, the mathematician is guessing.