A rectangle into a square

A garden bed measures 9 by 4. What side gives a square the same space?

The rule

The side is 6, because 6 × 6 = 36 = 9 × 4.

Squaring a shape means making a square with the same area.

Rule: Carry the width past the long side. Swing a semicircle on the whole length. Raise a perpendicular where the long side ends. That length is the square's side.

Why 6? The semicircle's radius is 13 ÷ 2 = 6.5. Its centre is the middle of the baseline. That is 6.5 from the left end. The 9-mark is 9 from the left end. So the gap is 9 − 6.5 = 2.5.

The radius, that gap and the perpendicular make a right triangle. The radius is the longest side. So the perpendicular squared is 6.5 × 6.5 − 2.5 × 2.5 = 42.25 − 6.25 = 36.

The Sulbasutras are old Indian books of drawing rules. They also turn rectangles into squares.

Different beds, same idea. Take 8 by 2. The radius is 10 ÷ 2 = 5. The gap is 8 − 5 = 3. So 5 × 5 − 3 × 3 = 16, and 4 × 4 = 16. The perpendicular squared always equals the bed's area.

In the lab, drag the 9-mark along the line. Or move the sliders. The perpendicular grows and shrinks with the bed. Try 7 by 5. The side is √35, about 5.92. Its square is 35 again.