Garfield’s proof of Pythagorean theorem

James Garfield, the 20th president of the United States, gave the following proof of the Pythagorean TheoremMathworldPlanetmathPlanetmath in 1876. Consider the following trapezoidMathworldPlanetmath (note that this picture is half of the diagram used in Bhaskara’s proof of the Pythagorean theorem (

Recall that the area of a trapezoid with two parallelMathworldPlanetmathPlanetmath sides (in this case, the left and right sides) s1 and s2 and height h is


So the area of the trapezoid above is


The area of the yellow triangle (and that of the blue triangle) is


while the area of the red triangle (also a right triangle) is


The two areas must be equal, so

(a+b)22 =2ab2+c22
a2+2ab+b22 =ab+c22
a2+2ab+b2 =2ab+c2
a2+b2 =c2
Title Garfield’s proof of Pythagorean theoremPlanetmathPlanetmathPlanetmath
Canonical name GarfieldsProofOfPythagoreanTheorem
Date of creation 2013-03-22 17:09:33
Last modified on 2013-03-22 17:09:33
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 10
Author rm50 (10146)
Entry type Proof
Classification msc 51-00