Square Area
A = l²P = 4·lD = l·√2The area of a square is the side squared. The perimeter is four times the side. The diagonal is the side multiplied by the square root of two, roughly 1.414.
It is the simplest polygon to calculate because a single measurement defines everything: knowing the side, every other quantity follows.
Among all rectangles of equal perimeter, the square encloses the greatest area. With 40 metres of fencing, a 10 by 10 square covers 100 square metres; a 5 by 15 rectangle covers only 75.
That is why the square shape appears in packaging, rooms and beds whenever the goal is to make the most of boundary material — wall, fence, frame.
The same property, taken to its limit, explains why the circle is more efficient still: among all shapes of equal perimeter, it encloses the largest possible area.
A square’s diagonal is irrational relative to its side: no exact fraction expresses that ratio. It was this discovery, attributed to the Pythagoreans, that revealed the existence of irrational numbers.
In practice the diagonal answers the fitting question: a square object 1 metre on a side needs 1.41 metres of clear opening to pass through diagonally.
It also serves to check squareness: in a square layout, if both diagonals measure the same, the corners are right angles.
Frequently asked questions
Measure both diagonals. If they are equal, the corners are at ninety degrees. It is the check used on site because it is fast and needs no instrument.
Because among rectangles of equal perimeter it encloses the greatest area. With a perimeter of 40, the square gives 100 of area and a 5 by 15 rectangle gives 75.
About 1.414 metres — the side times the square root of two. The value is irrational, so any decimal is an approximation.
Square the factor. From metres to centimetres the factor is 100, so the area is multiplied by 10,000.