Understand the square

A square is a quadrilateral with four equal sides and four right angles. These two conditions matter together: equal sides alone describe a rhombus, while right angles alone describe a rectangle.

Properties that make the calculations work

Its diagonals are equal, perpendicular and bisect each other. Each diagonal has length a√2. The square has four lines of reflection symmetry and rotational symmetry through quarter-turns.

Where the model is useful

Use side, area, diagonal or perimeter to size a square drawing. For floor layouts, area alone does not account for grout, offcuts, irregular walls or material tolerances.

Measure the right thing

A diagonal is longer than a side. Square units describe area; a number in cm² must not be interpreted as cm.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k². This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area; the unit selector relabels rather than converts entered numbers.

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Wolfram MathWorld: Square provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.