What every quadrilateral has in common
Every quadrilateral has four straight sides, four vertices (corners) and four interior angles that always add up to 360 degrees. That angle fact is worth checking early: cut any quadrilateral from paper, tear off the four corners and fit them together, and they make a full turn.
The different types are defined by extra rules about the sides and angles: which sides are parallel, which are equal, and which angles are right angles. The more rules a shape satisfies, the more special it is, and the more names it can carry at once.
The six types at a glance
| Shape | Sides | Angles | Parallel sides | Also counts as |
|---|---|---|---|---|
| Square | 4 equal | 4 right angles | 2 pairs | Rectangle, rhombus, parallelogram, kite |
| Rectangle | Opposite sides equal | 4 right angles | 2 pairs | Parallelogram |
| Rhombus | 4 equal | Opposite angles equal | 2 pairs | Parallelogram, kite |
| Parallelogram | Opposite sides equal | Opposite angles equal | 2 pairs | Nothing more general except quadrilateral |
| Trapezium (trapezoid) | No rule on lengths | No rule | Exactly 1 pair | Quadrilateral only |
| Kite | 2 pairs of adjacent equal sides | 1 pair of equal angles | Usually none | Quadrilateral only |
The quadrilaterals taught in primary and lower secondary school. Trapezium is the UK and Australian name; in the US the same shape is called a trapezoid.
Square
A square has four equal sides and four right angles. Because it meets every rule at once, it is the most special quadrilateral of all: it is a rectangle (four right angles), a rhombus (four equal sides), a parallelogram (two pairs of parallel sides) and even a kite (two pairs of equal adjacent sides).
Children often resist the idea that a square is a rectangle. The clearest explanation is that the rectangle rule only says four right angles and opposite sides equal. It never says the sides must be different lengths, so a square passes the test.
Rectangle
A rectangle has four right angles, with opposite sides equal and parallel. Its diagonals are equal in length and cut each other in half, which is a handy way to check whether a drawn shape really is a rectangle: measure both diagonals.
Rhombus
A rhombus has four equal sides, and its opposite angles are equal. Push a square over sideways and you get a rhombus: the sides stay equal, but the angles are no longer right angles. Its diagonals cross at right angles, and each diagonal cuts the other in half.
Many children call a rhombus a diamond. Diamond is fine as everyday language for a rhombus standing on a corner, but it is not a mathematical name, and a diamond turned to sit on a side is still a rhombus.
Parallelogram
A parallelogram has two pairs of parallel sides. Opposite sides are equal, opposite angles are equal, and the angles next to each other add up to 180 degrees. Rectangles, rhombuses and squares are all parallelograms with extra rules added.
Worked example: a parallelogram has one angle of 70 degrees. Its opposite angle is also 70 degrees, and the two remaining angles are each 180 minus 70, which is 110 degrees. Check: 70 + 70 + 110 + 110 = 360.
Trapezium (trapezoid)
A trapezium has exactly one pair of parallel sides. The parallel sides are usually called the bases, and the other two are the legs. If the legs are equal in length it is an isosceles trapezium, and its base angles are equal.
Naming differs by country. In the UK, Australia and New Zealand this shape is a trapezium. In the United States the same shape is a trapezoid, and trapezium there means a quadrilateral with no parallel sides at all. Worksheets and textbooks from a different country can quietly swap the two, so check which convention your curriculum uses.
Kite
A kite has two pairs of equal sides, but unlike a parallelogram the equal sides are next to each other rather than opposite. One pair of opposite angles is equal (the angles between the unequal sides), and the diagonals cross at right angles.
A rhombus is a kite where both pairs happen to be the same length, so all four sides are equal. That is why the table above lists rhombus and square as kites too.
How to teach the family tree
Teach the shapes as a hierarchy rather than a list of six unrelated names. Start with quadrilateral as the big family. Split into shapes with two pairs of parallel sides (parallelograms), one pair (trapeziums) and none (kites and irregular quadrilaterals). Then add rules within the parallelogram branch: four right angles makes a rectangle, four equal sides makes a rhombus, both together makes a square.
- Sorting cards: give mixed shape cards and ask children to sort by one rule at a time (parallel sides, right angles, equal sides) before combining rules.
- Always and sometimes: A square is always a rectangle. A rectangle is sometimes a square. A trapezium is never a parallelogram. Children justify each statement using the definitions.
- Mark the properties: on a printed shape, children mark equal sides with dashes, parallel sides with arrows and right angles with a small square. Reading the marks back gives the name.
- Draw to order: ask for a quadrilateral with exactly one right angle, or a parallelogram that is not a rectangle. The impossible requests (a trapezium with two pairs of parallel sides) are the best discussion starters.
Common mistakes
- Believing a shape can only have one name. A square really is a rectangle, a rhombus and a parallelogram at the same time.
- Judging by orientation: a rotated square is still a square, and a rhombus is not a different shape when it stands on a corner.
- Confusing trapezium and trapezoid across UK and US resources.
- Assuming a kite must look like a toy kite. Any quadrilateral with two pairs of equal adjacent sides is a kite, including very flat or very tall ones.