Square
A square is a regular quadrilateral with four equal sides and four right angles (90°).
Properties
A square is a special case of both a rectangle (all angles 90°) and a rhombus (all sides equal). Its diagonals are equal, perpendicular, and bisect each other.
Basic Formulas
S = a²
Area (side squared)
P = 4a
Perimeter (four times side length)
d = a√2
Diagonal (Pythagorean theorem)
R = (a/2)√2
Circumradius (radius of circumscribed circle)
r = a/2
Inradius (radius of inscribed circle)
Example
Given: side a = 6 cm
Area: S = 6² = 36 cm²
Perimeter: P = 4 · 6 = 24 cm
Diagonal: d = 6√2 ≈ 8.485 cm
Circumradius: R = 3√2 ≈ 4.243 cm
Inradius: r = 3 cm
Notes
The diagonal divides the square into two equal right isosceles triangles with angles 45°–45°–90°.
The square has the largest area among all quadrilaterals with the same perimeter.