9.1 Doubling a Square

A Question from the Sulba-Sutra Inside his Sulba-Sutra (around 800 BCE), a handbook of geometry written to help build fire-altars, the sage Baudhayana poses a question that sounds easier than it is:

Given a square, how do you draw a second square whose area is exactly twice as large?

The instinctive move is to double every side. See where that leads. Start from a square of side 1; doubling each side gives a square of side 2, whose area is

2 \times 2 = 4 \ \text{times the original}, \text{way too large!}

Doubling the side quadruples the area rather than doubling it. Baudhayana’s reply, in Verse 1.9, is far cleverer:

Baudhayana’s Insight The square drawn on the diagonal of a square has twice the area of that square.

So build the new square not on a side, but on the diagonal of the old one.

Why does the diagonal square come out at exactly twice the area? Slice the original square along a diagonal and you get two matching right triangles. Now sketch the larger, slanted square resting on that diagonal. Rule in the same up-down and left-right guide lines Baudhayana favoured, and you will count that this tilted square is built from four of those identical triangles.

original (2 triangles) on the diagonal (4 triangles)

Two triangles in the small square, four in the tilted one, and all of them congruent, so the diagonal square holds exactly twice the area. Run the same trick again and again, and you generate a chain of squares, each one double the last, made of 2, 4, 8, \dots little triangles.

Math Talk Why must the up-down and left-right sides of the small square, when extended, hit the corners of the tilted (dotted) square? Hint: inside a square, the line splitting a corner angle in half runs straight to the opposite vertex. Show that the small square’s horizontal and vertical sides bisect two angles of the big tilted square, so they have to land on its corners.

Try This: Doubling with Paper Cut out two matching paper squares. Keep the first one intact. Slice the second along both diagonals into four triangles (mark them 5, 6, 7, 8). Fit those four triangles snugly around the first square, they close around it to form a square of exactly double the area. You have just reconstructed Baudhayana’s idea by hand.