a rectangle
/ REK-tang-gul /
A rectangle is the shape of a door, a sheet of paper, a phone screen, or a brick face — a four-sided figure with four square (right-angle) corners. It is the everyday image of 'rectangular': straight, upright, with opposite sides matching.
Precisely, a rectangle is a parallelogram with one right angle — and that single right angle forces all four corners to be right angles (90 degrees), because in a parallelogram opposite angles are equal and consecutive angles are supplementary. Being a parallelogram, a rectangle has opposite sides parallel and equal and diagonals that bisect each other; its defining right angles then add one signature extra property: the two diagonals are equal in length. A rectangle whose sides are also all equal is a square; otherwise its length and width differ and it is sometimes called an oblong.
The rectangle is the 'equal-angles' branch of the parallelogram family, mirroring the rhombus's 'equal-sides' branch, with the square at their intersection. A clean reverse test: a parallelogram with equal diagonals must be a rectangle. The misconception to flag is that a rectangle is automatically a square — it is not; a square is a special rectangle (sides all equal), but most rectangles are longer than they are wide. Conversely every square is a rectangle, since a square has the four right angles a rectangle requires.
In rectangle ABCD all four angles are 90 degrees, opposite sides are equal, and the diagonals AC and BD are equal and bisect each other at the centre. If AB = 12 and BC = 5, each diagonal is sqrt(12^2 + 5^2) = 13, by the Pythagorean theorem on right triangle ABC.
Equal diagonals are the rectangle's signature beyond being a parallelogram.
Every square is a rectangle, but not every rectangle is a square — a rectangle only needs four right angles, while a square also needs all sides equal.