Square Calculator

Enter the side length to compute area, perimeter, diagonal, circumradius, and inradius of a square.

Enter Side Length

Results

Area
25
square units
Perimeter20
Diagonal7.071068
Circumradius (R)3.535534
Inradius (r)2.5

Step-by-Step Solution

Understanding Squares

A square is a regular quadrilateral with four equal sides and four right angles (90 degrees). It is one of the most fundamental shapes in geometry. Every square is simultaneously a rectangle, a rhombus, and a parallelogram, making it a highly symmetric figure.

Square Formulas

Area

The space enclosed within the square.

A = s²

Perimeter

The total length of all four sides.

P = 4s

Diagonal

The distance between two opposite corners.

d = s√2

Circumradius

Radius of the circumscribed circle passing through all vertices.

R = s√2 / 2

Inradius

Radius of the inscribed circle tangent to all four sides.

r = s / 2

Properties of a Square

  • All four sides are equal in length.
  • All four interior angles are 90 degrees.
  • The diagonals are equal in length and bisect each other at right angles.
  • Each diagonal divides the square into two congruent isosceles right triangles.
  • The diagonals bisect the vertex angles (each diagonal creates two 45-degree angles).
  • A square has four lines of symmetry and rotational symmetry of order 4.

Practical Applications

Squares appear everywhere in architecture, engineering, tiling, and design. Floor tiles, chessboards, pixels on a screen, and building floor plans frequently use square geometry. Understanding square measurements is essential for calculating material needs, area coverage, and structural planning.