A square with the same ground

A courtyard is 9 paces by 4. You want a square with the same ground, meaning the same area. What side? Slide the length.

The ground, and the side

The side is 6 paces. The rectangle covers 9 × 4 = 36 square paces. A square of side s covers s × s. So s × s = 36, and s = 6.

Try a plot 9 by 5. It covers 45. Now 6 × 6 = 36 and 7 × 7 = 49. The side is between 6 and 7, because 36 < 45 < 49. It is not a whole number.

Same ground, different way round

The rectangle's way round, or perimeter, is 2 × (9 + 4) = 26 paces. The square's way round is 4 × 6 = 24 paces. Equal ground does not mean an equal way round.

A corridor 16 by 1 covers 16 square paces. So does a 4 by 4 square. Their ways round are 34 and 16.

Building it

A square needs four equal sides and four right angles. Each right angle is 90°. Postulate 4 says every right angle equals every other. So all four corners of the square are equal angles. A compass copies one length four times. So the four sides are equal.

Rule: The side of the equal square, times itself, gives the area. Equal ground does not mean an equal way round.