Square Calculator

Calculate square properties: area, perimeter, diagonal, and side length. Visualize squares and understand geometric formulas.

Calculate from Side
Calculate from Area
Calculate from Perimeter
Calculate from Diagonal

Square Formulas:

Area: A = s²

Perimeter: P = 4s

Side: s = √A

Diagonal: d = s√2

Length of one side of the square
Calculating...

Square Geometry

A square is a regular quadrilateral with four equal sides and four right angles (90°). It is a special case of a rectangle and a rhombus. All squares are similar to each other.

Key Properties of a Square:

  • All four sides are equal in length
  • All four interior angles are right angles (90°)
  • Opposite sides are parallel
  • The diagonals bisect each other at right angles
  • The diagonals are equal in length
  • It has four lines of symmetry
  • It has rotational symmetry of order 4

Square Formulas

Property Formula Description
Area A = s² Side length squared
Perimeter P = 4s Four times the side length
Diagonal d = s√2 Side length times square root of 2
Side from Area s = √A Square root of area
Side from Perimeter s = P/4 Perimeter divided by 4
Side from Diagonal s = d/√2 Diagonal divided by square root of 2
Inradius (r) r = s/2 Radius of inscribed circle
Circumradius (R) R = d/2 = s/√2 Radius of circumscribed circle

How to Calculate Square Properties

1

From Side Length: If you know the side length (s), calculate area (A = s²), perimeter (P = 4s), and diagonal (d = s√2).

2

From Area: If you know the area (A), calculate side length (s = √A), then perimeter (P = 4s) and diagonal (d = s√2).

3

From Perimeter: If you know the perimeter (P), calculate side length (s = P/4), then area (A = s²) and diagonal (d = s√2).

4

From Diagonal: If you know the diagonal (d), calculate side length (s = d/√2), then area (A = s²) and perimeter (P = 4s).

Applications of Squares

  • Architecture: Square floor plans, tiles, and structural elements
  • Art & Design: Grid systems, layouts, and geometric patterns
  • Mathematics: Basis for coordinate systems (Cartesian coordinates)
  • Engineering: Square cross-sections in beams and structural components
  • Computer Graphics: Pixels and digital imaging are based on square grids
  • Land Measurement: Square units for area measurement

Interesting Square Facts:

  • A square is the most symmetrical quadrilateral
  • The ratio of a square's diagonal to its side is √2 (approximately 1.414)
  • Squares tessellate perfectly (can cover a plane without gaps)
  • A square is both a rectangle and a rhombus
  • The ancient Greeks discovered that √2 is irrational while studying squares

Frequently Asked Questions

All squares are rectangles, but not all rectangles are squares. A rectangle has four right angles with opposite sides equal, while a square has four right angles with all four sides equal. A square is a special case of a rectangle where all sides have equal length.

This comes from the Pythagorean theorem. The diagonal forms a right triangle with two sides of the square. So d² = s² + s² = 2s², therefore d = √(2s²) = s√2.

Mathematically, a square with side length zero is a degenerate square (a point). It has zero area, zero perimeter, and zero diagonal. In practical applications, squares typically have positive side lengths.

Since the diagonal d = s√2, you can find the side length by dividing the diagonal by √2: s = d/√2. Alternatively, s = d × (√2/2) ≈ d × 0.7071.

For a square with side s, area A = s² and perimeter P = 4s. You can express one in terms of the other: s = √A = P/4, so A = (P/4)² = P²/16, and P = 4√A.