What does this square calculator do?

This calculator combines algebra and geometry: it computes the algebraic square x² and √x for a real number, and it also converts between side length, area, perimeter, and diagonal of a square. You can start from any one of these four geometric quantities and the tool derives the others, with formulas explained step by step.

Which units can I use for the square geometry?

You can work with common metric and imperial units such as millimetres, centimetres, metres, inches and feet. The tool keeps all geometric results in the same base unit: linear values in the chosen unit and the area in its squared version (for example m², cm², in²).

Can I enter negative values?

For the algebraic square x², negative inputs are allowed and treated in the usual way (for example, (−3)² = 9). For the geometry part, side, area, perimeter, and diagonal must be strictly positive; negative or zero values are rejected because they do not correspond to a real square.

What is the difference between x² and the area of a square?

The algebraic square x² is a purely numerical operation: you multiply a number by itself, without any context. The area of a square uses the same mathematical operation but with units attached: if the side is s metres, then the area is s² square metres. The calculator distinguishes clearly between these two use cases.

Square Calculator – x², Area, Perimeter & Diagonal

A professional-grade square calculator that covers both algebra (x² and √x) and geometry of a square (side, area, perimeter, diagonal). Enter any known quantity and see all others computed with clearly documented formulas.

Algebraic square x² Square area & side Perimeter & diagonal

Display precision

Internal computations always use full double precision. The selector only affects how results are rounded for display.

1. Algebraic square calculator (x² and √x)

This card treats the square as a pure algebraic operation. Enter a real number x to compute its square \(x^2\) and, when defined, its principal square root \(\sqrt{x}\).

Number x

You can use either “.” or “,” as decimal separator. Scientific notation (e.g. 1e6) is also accepted.

2. Square geometry calculator (side, area, perimeter, diagonal)

Here the square is a geometric shape with side \(s\). This calculator converts between: side \(s\), area \(A\), perimeter \(P\), and diagonal \(d\). Provide any one of these, choose a unit, and the others are computed.

I know this quantity

Side s Area A Perimeter P Diagonal d

Unit

Perimeter and diagonal use the same unit; area uses its squared version (e.g. m²).

Side s

Must be a positive value. Use “.” or “,” for decimals. Scientific notation is accepted.

Algebraic square vs. square as a shape

In algebra, the square of a number is simply \[ x^2 = x \cdot x. \] This operation is defined for all real numbers, and negative inputs are perfectly valid: for example, \((-3)^2 = 9\).

In geometry, a square is a polygon with four equal sides and four right angles. If its side length is \(s\), then:

\[ A = s^2 \quad \text{(area)}, \qquad P = 4s \quad \text{(perimeter)}, \qquad d = s\sqrt{2} \quad \text{(diagonal)}. \]

Square geometry formulas in detail

Suppose you know any one of the four quantities \((s, A, P, d)\). You can recover the others as follows:

From side s
\[ A = s^2, \quad P = 4s, \quad d = s\sqrt{2}. \] From area A
\[ s = \sqrt{A}, \quad P = 4\sqrt{A}, \quad d = \sqrt{2A}. \] From perimeter P
\[ s = \frac{P}{4}, \quad A = \left(\frac{P}{4}\right)^2, \quad d = \frac{P}{4}\sqrt{2}. \] From diagonal d
\[ s = \frac{d}{\sqrt{2}}, \quad A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}, \quad P = 4\frac{d}{\sqrt{2}}. \]

Units: from length to area

If the side is measured in metres, the area is measured in square metres: \[ [s] = \text{m} \quad \Rightarrow \quad [A] = \text{m}^2. \] The same applies to any unit (centimetres, inches, feet, etc.). The calculator automatically keeps the area unit consistent with your choice of length unit.

FAQ – Using the square calculator

Can I use the tool for foundations, slabs, or tiles?

Yes. If you know the side or the area of a square footing, slab, tile, or panel, the geometry card gives you all derived quantities. For example, if a square tile has side 30 cm, the calculator returns an area of 900 cm² (= 0.09 m²) and a diagonal of about 42.43 cm.

Does the algebraic x² calculator understand units?

No. The algebraic section treats x as a dimensionless number. If you are working with physical quantities, use the geometry card so that units are tracked consistently.

How many decimal places should I keep?

For engineering and construction, two to three decimals often suffice for lengths in metres (e.g. 0.001 m = 1 mm). For academic work, you can switch to 4–6 decimals. The precision selector lets you adapt rounding to your context without sacrificing internal accuracy.