AJ Designer

Square Calculator

Area equals side squared

Solution

—
Share:

Calculate Square Area from Side Length

Use this form when the side length is known and you need the enclosed area — square plots, tiles, square tables.

A = s²

Calculate Square Perimeter from Side Length

Use this form to compute the perimeter when the side length is known — fencing or trim around a square area.

P = 4s

Calculate Square Diagonal from Side Length

Use this form to compute the diagonal — fitting a square into a circular opening, or finding the longest distance inside the square.

d = s√2

Calculate Square Side from Area

Use this rearrangement when the area is known and you need the side length — sizing a square plot to a target area.

s = √A

How It Works

This square calculator solves the basic square formulas: area A = s², perimeter P = 4s, and diagonal d = s√2. Pick the unknown with the solve-for toggle, enter the remaining value in any supported length or area unit, and the calculator converts to SI internally before computing. All four quantities (s, A, P, d) are always shown together so you get the complete picture from a single page.

Example Problem

A square tile has a 4 cm side length. What are its area, perimeter, and diagonal?

  1. Knowns: s = 4 cm
  2. Area: A = s² = 4² = 16 cm²
  3. Perimeter: P = 4s = 4 · 4 = 16 cm
  4. Diagonal: d = s√2 = 4√2 ≈ 5.657 cm
  5. Sanity check (inverse): from A = 16, s = √16 = 4 cm, recovering the original side.

Square is the only rectangle where area and perimeter have the same numeric value at s = 4 (16 = 16). For s < 4 perimeter exceeds area; for s > 4 area dominates.

When to Use Each Variable

  • Solve for Area — when the side length is known and you need the surface area.
  • Solve for Perimeter — when you need the total boundary length — fencing, trim, packaging.
  • Solve for Diagonal — when you need the corner-to-corner distance — fitting into a circular space or bracing.
  • Solve for Side — when the area is known and you need the side length.

Key Concepts

A square is a regular quadrilateral — four equal sides and four right angles. A single number, the side length s, fully determines its geometry. Every property (area, perimeter, diagonal) derives from s. The diagonal d = s√2 follows directly from the Pythagorean theorem applied to two adjacent sides. Squares are the foundational unit for tiling, grid systems, and area measurement.

Applications

  • Tiling and flooring: count or size square tiles for a given floor area
  • Land surveying: compute area of square parcels
  • Packaging: fit square products into square or rectangular cartons
  • Geometry instruction: foundational shape for area, perimeter, and Pythagorean-theorem problems

Common Mistakes

  • Confusing area (s², square units) with perimeter (4s, linear units)
  • Forgetting that the diagonal is s√2, not 2s
  • Using s² for the perimeter or 4s for the area — the two have similar-looking numbers but different units and meanings

Frequently Asked Questions

How do you calculate the area of a square?

Square the side length: A = s². For a square with s = 4 cm, A = 16 cm².

What is the formula for the perimeter of a square?

P = 4s. Multiply the side length by 4. For s = 4 cm, P = 16 cm.

How do you find the diagonal of a square?

d = s√2. The diagonal of a square is its side length multiplied by √2 (approximately 1.4142). For s = 4 cm, d ≈ 5.657 cm.

How do you find the side of a square given its area?

Take the square root: s = √A. For A = 16 cm², s = √16 = 4 cm.

Why is the diagonal of a square s√2?

The diagonal cuts the square into two right triangles with legs s and s. By the Pythagorean theorem, the hypotenuse (the diagonal) is √(s² + s²) = √(2s²) = s√2.

What is the difference between a square and a rectangle?

A square is a rectangle where all four sides are equal. Every square is a rectangle, but only the rectangles with l = w are squares. Squares have a single side dimension; rectangles need two.

What is the area of a 10×10 square?

10² = 100 square units. The unit depends on the side unit — 10 cm gives 100 cm², 10 inches gives 100 in², 10 m gives 100 m².

How is the side length of a 50 m² square plot calculated?

Take the square root of the area: s = √50 ≈ 7.071 m. So a 50 m² square plot is about 7.07 m × 7.07 m.

Reference: Weisstein, Eric W. "Square." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/Square.html

Worked Examples

Tiling

How many 30 cm tiles cover a 3 m × 3 m floor?

A square floor measures 3 m × 3 m. Compute the area and divide by the tile area to count tiles.

  • Floor area: A = 3² = 9 m²
  • Tile area: 0.30² = 0.09 m²
  • Tile count: 9 / 0.09 = 100 tiles

100 tiles cover the floor exactly

Add waste factor (5–10%) when ordering tiles to account for cuts.

Land

What is the perimeter of a 50 m × 50 m square lot?

A square land parcel measures 50 m on each side. Find the perimeter to estimate fencing.

  • Knowns: s = 50 m
  • Formula: P = 4s
  • P = 4 · 50 = 200 m

Perimeter = 200 m (about 656 ft)

Real-world fencing typically includes a gate and corner posts — budget extra material.

Sizing

What is the side of a 100 m² square plot?

A square plot has an area of exactly 100 m². Find the side length.

  • Knowns: A = 100 m²
  • Formula: s = √A
  • s = √100 = 10 m

Side ≈ 10 m (10 m × 10 m square)

Most square land plots are not perfect squares in practice — survey marks usually have small angular and dimensional tolerances.

Square Formulas

All square properties follow from a single dimension: the side length s. Area, perimeter, and diagonal all derive from it:

Area equals side squaredPerimeter equals four sDiagonal equals s times the square root of 2Side equals the square root of area
Square with side s and diagonal dsd

Where:

  • A — area (m², ft², in², etc.)
  • P — perimeter (m, ft, in)
  • d — diagonal length
  • s — side length (m, cm, in, ft, yd)

Related Calculators

Related Sites