Why does Baudhāyana’s method work?

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Homi · Accepted Answer

The diagram shows a visual proof of the Pythagorean theorem.

Step 1 — Identifying the squares

We start with two squares of different sizes. Let the side length of the smaller square be $a$ . Its area is $a \times a = a^2$ . Let the side length of the larger square be $b$ . Its area is $b \times b = b^2$ .

Diagram 1

Step 2 — Forming a right-angled triangle

These two squares are then arranged. One side of the square with area $a^2$ forms one leg of a right-angled triangle. One side of the square with area $b^2$ forms the other leg of the right-angled triangle. These two sides meet at a right angle (an angle of 90 degrees). Let the length of the hypotenuse (the side opposite the right angle) of this triangle be $c$ .

Diagram 2

Step 3 — Constructing the third square

A third square is constructed. This third square is built on the hypotenuse of the right-angled triangle. The diagram clearly labels the area of this third square as "Area A + Area B". This means the area of the square on the hypotenuse is the sum of the areas of the other two squares.

$\text{Area of square on hypotenuse} = \text{Area A} + \text{Area B}$

$\text{Area of square on hypotenuse} = a^2 + b^2$

We know that the area of a square with side length $c$ is $c^2$ . So, the diagram visually demonstrates that:

$\boxed{c^2 = a^2 + b^2}$

Step 4 — Baudhāyana's method

Baudhāyana's method, as described in his Sulbasutra, is a geometric way to understand this relationship. It states that the square constructed on the diagonal (hypotenuse) of a rectangle has an area equal to the sum of the areas of the squares constructed on its sides (legs). The diagram perfectly illustrates this principle. It shows that the area of the square built on the hypotenuse is exactly equal to the combined areas of the squares built on the other two sides. This is the Pythagorean theorem (also known as Baudhāyana's theorem in India). Therefore, Baudhāyana's method works because it provides a geometric construction that proves the fundamental relationship between the sides of a right-angled triangle.

Answer

Baudhāyana’s method works because it geometrically demonstrates the Pythagorean theorem. It shows that when squares are constructed on the two shorter sides (legs) of a right-angled triangle, the sum of their areas is equal to the area of the square constructed on the longest side (hypotenuse). This means if the legs have lengths $a$ and $b$ , and the hypotenuse has length $c$ , then $a^2 + b^2 = c^2$ .

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