Garfield's trapezoid
James A. Garfield's proof of the Pythagorean theorem with a trapezoid made of three right triangles.
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Step 1 of 5
Garfield's trapezoid
- A trapezoid with parallel sides a and b, a + b apart. Its area is half the sum of the parallel sides times the distance between them: ½(a + b)(a + b) = ½(a + b)².
- The lower-left triangle is the right triangle with legs a and b. Its area is ½ab.
- The lower-right triangle is a copy of it, turned: legs b and a, area ½ab.
- The middle triangle has two sides c, the two hypotenuses. The angle between them is a right angle: with the two acute angles of the right triangle, which add up to 90°, it makes a straight line. Its area is ½c².
- The three triangles fill the trapezoid: ½(a + b)² = ½ab + ½ab + ½c². Doubled: a² + 2ab + b² = 2ab + c², so a² + b² = c².
James A. Garfield, New-England Journal of Education 3 (1876), p. 161; redrawn
Where this proof comes from
James A. Garfield found this proof in 1876, while he was a congressman from Ohio. It was printed that year in the New-England Journal of Education (volume 3, page 161). In 1881 he became the 20th president of the United States.
Loomis, The Pythagorean Proposition (1927), geometric proof 157, p. 185. Free scan: HathiTrust