Classifying Quadrilaterals

A quadrilateral is any closed shape made from four straight sides. It has four corners, four sides and four angles, and once you label the corners you can talk about every part of it. This one is quadrilateral ABCD, named by walking round the outline from A to B to C to D and back to A. Nothing about it is special, so it does not get a fancier name than that. Other quadrilaterals do earn special names, and they earn them by having a particular property: sides that are parallel, sides that are equal in length, or corners that are right angles. Each special name packs several facts into one word.

The first property to look for is parallel sides, and the arrowheads on a diagram tell you which sides those are. If a quadrilateral has exactly one pair of parallel sides, it is a trapezium. Here the top and the bottom run in the same direction and would never meet, but the two slanting sides clearly would, so only one pair qualifies. If a quadrilateral has two pairs of parallel sides, it is a parallelogram, and the second pair is marked with double arrowheads to keep the pairs apart. A parallelogram comes with a bonus fact: because its opposite sides are parallel, they are also equal in length.

Three more names come from adding an extra condition to a parallelogram. A rectangle is a parallelogram whose four corners are all right angles, so its opposite sides are equal but the long sides need not match the short ones. A rhombus is a parallelogram whose four sides are all the same length, and its corners do not have to be right angles at all, which is why it can lean over. A square is the fussiest of the lot, because it demands both conditions at once: four equal sides and four right angles. Read the marks on a diagram and the name follows straight away.

A kite is the odd one out, because its equal sides sit next to each other rather than opposite each other. Here the two short sides meet at the top and both carry one tick, while the two long sides meet at the bottom and both carry two ticks. Nothing in a kite is parallel. That is exactly what separates it from a parallelogram, where the equal sides are the opposite ones. And plenty of quadrilaterals have no equal sides, no parallel sides and no right angles at all. Those keep the plain name quadrilateral, and calling one of them simply a quadrilateral is a complete and correct answer.

The special names are not six separate boxes, because a shape keeps every name it qualifies for. Start at the top with quadrilateral, which covers everything. A parallelogram is a quadrilateral with two pairs of parallel sides. A rectangle is a parallelogram with right angles, and a rhombus is a parallelogram with four equal sides. A square sits at the bottom, satisfying every condition above it, so a square is also a rectangle, also a rhombus and also a parallelogram. Reading downwards always works, but reading upwards does not. Most rectangles are not squares, and most parallelograms are not rhombuses.