Using the constructions you have now seen, how would you construct a square whose area is triple the area of a given square? Five times the area of a given square? (Baudhāyana's Śulba-Sūtra, Verse 1.10)
To construct a square with triple the area:
Step 1: Construct a square with double the area using the diagonal method.
Step 2: Place the original square adjacent to the doubled square to form an L-shape.
Step 3: Complete this into a right triangle with legs equal to the side of the original square and the side of the doubled square (which is the diagonal of the original square).
Step 4: The square on the hypotenuse of this right triangle will have area equal to a² + 2a² = 3a², which is triple the original area.
Why it works: Baudhāyana's method combines two different squares by constructing a right triangle whose perpendicular sides equal the side lengths of the two squares. The square on the hypotenuse has area equal to the sum of the two original squares. For triple area, we add a² + 2a² = 3a², requiring a right triangle with legs a and a (the diagonal of the original square).
To construct a square with five times the area:
Method 1: Combine a square of area a² with a square of area 4a² (which is the double of a doubled square). The hypotenuse of a right triangle with these as legs gives a square of area 5a².
Method 2: Use a right triangle with legs a and 2a to get hypotenuse a, giving a square of area 5a².
Why it works: For five times the area, we add a² + 4a² = 5a², requiring legs a and 2a.
Explanation
The chapter explains Baudhāyana's method for combining two different squares by constructing a right triangle whose perpendicular sides equal the sidelengths of the two squares. The square on the hypotenuse has area equal to the sum of the two original squares. For triple area, we need to add a² + 2a² = 3a². This requires a right triangle with legs a and a (the diagonal of the original square). For five times, we add a² + 4a² = 5a², requiring legs a and 2a.
Solution Steps
Construct a square with double the area using the diagonal method.
Place the original square adjacent to the doubled square to form an L-shape.
Complete this into a right triangle with legs equal to the side of the original square and the side of the doubled square (which is the diagonal of the original square).
The square on the hypotenuse will have area a² + 2a² = 3a².
For five times the area: Combine a square of area a² with a square of area 4a².
The hypotenuse of a right triangle with these as legs gives a square of area 5a².
Alternatively, use a right triangle with legs a and 2a to get hypotenuse a, giving a square of area 5a².