Let the original square have a side length of s. If we double the side length to 2s, the new area becomes (2s)² = 4s², which is 4 times the original area rather than double.

Constructing a square whose side length equals the diagonal d gives an area of d² = 2s². Since the original area is s², the square constructed on the diagonal has exactly 2 times the area of the original square.

Constructing a square whose side length equals the diagonal d gives an area of d² = 2s². Since the original area is s², the square constructed on the diagonal has exactly 2 times the area of the original square.

2.1 Doubling a Square In Baudhāyana's Śulba-Sūtra (c. 800 BCE), Baudhāyana considers the following question: How can one construct a square having double the area of a given square? A first guess might be to simply double the length of each side of the square. Will this new square have double the area of the original square? It's not hard to see that the new square will have area $2 \times 2 = 4$ times the area of the original square: [Diagrams shown: 1. A small square with side markings indicating equal side lengths. 2. A larger square divided into a $2 \times 2$ grid of 4 smaller squares, showing dashed lines dividing it into quadrants, each quadrant having side lengths equal to the original square (indicated by hash marks).] So how can one make a square that has double the area? Baudhāyana in his Śulba-Sūtra (Verse 1.9) gave an elegant answer to this question: "The diagonal of a square produces a square of double the area of the original square."

Answer: Doubling the side length produces a square with 4 times the area. To construct a square with double the area, construct the square using the diagonal of the original square as its side length, since $d^2 = 2s^2$ .

Step-by-step solution

Step 1: Analyze the effect of doubling side length

Let the original square have a side length of $s$ . If we double the side length to $2s$ , the new area becomes $(2s)^2 = 4s^2$ , which is $4$ times the original area rather than double.

Step 2: State Baudhāyana's Sulba-Sutra principle

Baudhāyana observed in verse 1.9 of the *Śulba-Sūtra* that the diagonal of a square produces a square with double the area. By the Baudhāyana-Pythagoras theorem, the diagonal $d$ of a square with side $s$ satisfies $d^2 = s^2 + s^2 = 2s^2$ .

Step 3: Verify the area of the square on the diagonal

Constructing a square whose side length equals the diagonal $d$ gives an area of $d^2 = 2s^2$ . Since the original area is $s^2$ , the square constructed on the diagonal has exactly $2$ times the area of the original square.

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