Square

A square is a regular quadrilateral with four equal sides and four right angles (90°).

d a a 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.

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