Quadrilaterals Overview and Relationships
A quadrilateral is a polygon with four sides, and the special types (trapezoid, parallelogram, rectangle, rhombus, square and kite) are related by inclusion: every square is a rectangle, every rectangle is a parallelogram, and so on. In brief: the interior angles of any quadrilateral add up to 360°. A parallelogram has two pairs of parallel sides, a rectangle is a parallelogram with four right angles, a rhombus is a parallelogram with four equal sides and a square is both a rectangle and a rhombus.
Many shapes in everyday life are quadrilaterals, and their names are easy to confuse. The key to order is the family tree of definitions: each special quadrilateral has all the properties of the types above it. This guide explains the general quadrilateral, the angle sum, the family tree, how to identify a quadrilateral from its properties and how the diagonals help, with a table that compares the properties.
What is a quadrilateral?
A quadrilateral has four vertices, four sides and two diagonals. It is convex if both diagonals lie inside it, and concave if one vertex points inward. The sum of the interior angles of any simple quadrilateral is 360°: a diagonal splits it into two triangles, each with an angle sum of 180°. The perimeter is the sum of the four sides. In a quadrilateral with three angles of 80°, 95° and 110°, the fourth is 360° − 285° = 75°.
A quadrilateral is determined by more data than a triangle: four sides do not fix its shape, since a flexible frame of four rods can change its angles. A diagonal makes it rigid by splitting it into two triangles.
How are the special quadrilaterals related?
- Trapezoid (trapezium): at least one pair of parallel sides.
- Parallelogram: two pairs of parallel sides. It is a trapezoid in the inclusive definition.
- Rectangle: a parallelogram with four right angles.
- Rhombus: a parallelogram with four equal sides.
- Square: a rectangle that is also a rhombus.
- Kite: two pairs of adjacent equal sides. A rhombus is a kite, and so is a square.
The family tree: quadrilateral ⊃ trapezoid ⊃ parallelogram ⊃ {rectangle, rhombus} ⊃ square. The kite is another branch: quadrilateral ⊃ kite ⊃ rhombus ⊃ square. Some textbooks define a trapezoid as having exactly one pair of parallel sides, which excludes parallelograms from trapezoids. Both conventions exist, so check which one your course uses.
How do you identify a quadrilateral?
- Check for parallel sides: none (general or kite), one pair (trapezoid) or two pairs (parallelogram family).
- For the parallelogram family check the sides and the angles: all sides equal makes a rhombus, all angles right makes a rectangle, both makes a square.
- Check for adjacent equal sides to find a kite.
- Use the most specific name that fits.
Example 1: coordinates
A(0, 0), B(4, 0), C(6, 3), D(2, 3). Slopes: AB = 0, BC = 3/2, CD = 0, DA = 3/2. Two pairs of parallel sides, so a parallelogram. Sides: AB = 4, BC = √13, CD = 4, DA = √13. Unequal adjacent sides, so not a rhombus. Not a rectangle, as the sides are not perpendicular (slopes 0 and 3/2).
Example 2: from angles
A quadrilateral with angles 60°, 120°, 60°, 120° in this order has opposite angles equal and consecutive angles adding up to 180°, so the sides are parallel and it is a parallelogram.
Properties of quadrilaterals compared
| Property | Parallelogram | Rectangle | Rhombus | Square | Kite |
|---|---|---|---|---|---|
| Opposite sides parallel | Y | Y | Y | Y | N |
| Opposite sides equal | Y | Y | Y | Y | N |
| All sides equal | N | N | Y | Y | N |
| All angles 90° | N | Y | N | Y | N |
| Diagonals bisect each other | Y | Y | Y | Y | N |
| Diagonals equal | N | Y | N | Y | N |
| Diagonals perpendicular | N | N | Y | Y | Y |
What do the diagonals tell you?
The diagonals reveal the type of a quadrilateral. If the diagonals bisect each other, it is a parallelogram. If they are also equal, it is a rectangle, and if they are perpendicular as well, it is a rhombus (bisecting diagonals that are perpendicular). Equal, perpendicular and bisecting diagonals give a square. In a kite, one diagonal is the perpendicular bisector of the other.
Example 3
Vertices A(0, 0), B(3, 1), C(4, 4), D(1, 3). The midpoint of AC is (2, 2) and of BD is (2, 2). The diagonals bisect each other, so ABCD is a parallelogram. The slope of AC is 1, and the slope of BD, from (3, 1) to (1, 3), is (3 − 1)/(1 − 3) = −1. The product 1 · (−1) = −1, so the diagonals are perpendicular. The sides AB = √10 and BC = √10, so it is a rhombus. The diagonals AC = √32 and BD = √8 are unequal, so it is not a square.
Where do the names come from?
The word quadrilateral comes from Latin quadri (four) and latus (side). Rhombus is from a Greek word for a spinning top, and trapezoid comes from the Greek word for a table. Euclid classified quadrilaterals into square, oblong, rhombus, rhomboid and trapezia in Book I of the Elements. The names parallelogram and kite for the shapes we know today developed later, and the meaning of “trapezium” differs between British and American English.
Where are quadrilaterals used today?
Rectangles are everywhere: screens, books and rooms. Parallelograms appear in mechanisms such as scissor lifts and the linkages of folding tables, because opposite sides remain parallel as the angle changes. Kites are used as flying kites and in design, and trapezoids appear in bridges, roofs and the cross-section of canals.
Which mistakes are common?
Treating the categories as separate
A square is also a rectangle and a rhombus. Use the most specific name, but remember the inclusion.
Assuming equal sides imply a rhombus without checking all four
A kite can have two pairs of equal sides without being a rhombus. In a rhombus all four sides are equal.
Forgetting the convention for trapezoids
Whether a parallelogram counts as a trapezoid depends on the definition in use.
Assuming equal diagonals imply a rectangle
An isosceles trapezoid also has equal diagonals. A rectangle needs diagonals that are equal and bisect each other.