Look a little closer.

Turn two lengths into a complete picture. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

Equal perimeter does not imply equal area: different rectangles can use the same boundary length.

The diagonals are equal and bisect one another. A square is a rectangle whose adjacent sides happen to be equal. Unless it is a square, the diagonals are not perpendicular.

02 / A different perspective

Among rectangles of a fixed perimeter, the square has the greatest area.

A rectangle is a quadrilateral with four right angles. Opposite sides are equal and parallel. The two adjacent side lengths determine its area, perimeter and diagonal.

03 / A connection to explore

The diagonal follows the Pythagorean theorem.

Measure a room outline, a page or a screen using a common unit. Inverse calculators recover a missing side from area, perimeter or diagonal when one side is already known.

Try the idea with numbers

Open the length and width calculator, note its outputs, then multiply each length input by two. Area should increase by four and every reported length should double, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Rectangle at Wolfram MathWorld.