What procedure would you use to square a given triangle? Here the task is to construct a square whose area equals the area of a given triangle. How would you proceed?                                                                       [Page No. 142]

The construction can be carried out in two area-preserving steps.

Step 1: Convert the triangle into an equal-area rectangle

Let the triangle have base b and height h. Join the midpoints of the two sides that meet at the vertex opposite the base. This segment is parallel to the base and lies halfway up the height of the triangle.

Cut along this segment and move the smaller top triangle beside the lower trapezium. The pieces fit together to form a rectangle whose,

$$\text{length} = b \text{ and breadth} = \frac{h}{2}$$

Since the area is unchanged by cutting and rearranging, the rectangle has the same area as the original triangle

$$\text{Area of rectangle} = b \times \frac{h}{2} = \frac{1}{2} bh.$$

Thus, the triangle has been transformed into an equal-area rectangle.

Step 2: Convert the rectangle into an equal-area square

Now, apply Baudhāyana's construction for squaring a rectangle. This construction produces a square whose area is exactly equal to that of the rectangle.

Since the rectangle and the original triangle have the same area, the resulting square also has the same area as the original triangle.

Therefore, the process is Triangle → Equal-area Rectangle → Equal-area Square.

The first step uses a midpoint cut and rearrangement, while the second uses the rectangle-to-square construction. Hence a square equal in area to a given triangle can be constructed.