When I first heard the Pythagorean Formula, I wondered where did Pythagoras get the idea of that relationship? The teacher showed a right triangle and drew squares upon each side. That was so unintuitive to me. It was a deduction after the fact. It didn’t motivate my understanding of the formula.

Years later, I stumbled across a construction that made much more sense. Consider a square. Divide a side anywhere into two parts, named a and b. Then from the first a draw a straight line to point a on the adjacent side. Do the same to the third side, the fourth side, then close the inscribed square by drawing a straight line from the fourth side to the original point a.

Figure 1 shows the original square of side a+b and the inscribed square of side c.

Area of original square = Area of inscribed square + 4* identical right triangles

(a + b)^{2} = c^{2 }+ 4(½ ab)

a^{2 }+ 2ab + b^{2} = c^{2 }+ 2 ab

a^{2} + b^{2} = c^{2}

Perhaps some teachers use the inscribed square to explain the Pythagorean Theorem. I wish mine had.

Tabulators separate ballots into piles for each presidential candidate during a recount in Dane County (Wis.) on December 1, 2016 in Madison, Wisconsin. (Getty)