What is a Quadrangle? — Definition, Shapes, & Examples
Quadrangle definition
A quadrangle is a four-sided, plane figure that closes in a space. Nine quadrangle shapes have names: trapezium, trapezoid, isosceles trapezoid, parallelogram, rhombus, rectangle, square, kite, and dart. Quadrangles can be concave or convex, simple or complex, and can also be called quadrilaterals.
Quadrangle meaning
The word quadrangle is formed from two-word parts: the prefix quad–, meaning four, and –angle.
Quadrilateral and quadrangle are synonyms in geometry. Quadrilateral is also Latin, but it means “four sides.” A very different word, tetragon, comes from Greek origins but also means a four-sided shape.
Quadrangle shape
A quadrangle shape is a four-sided polygon that has four interior angles. The sides are straight and connect to close in an area.
Anytime you draw a shape on a flat sheet of paper that has these properties, you have drawn a quadrangle:
Four points connected by four straight sides
Four interior angles
Closed area
Quadrangle angles
All convex quadrangles have four interior angles that range in measurement between 0° and 180°. Concave quadrangles will have one interior angle that measures greater than 180°. The sum of the interior angles for all quadrangles is always 360°.
In quadrangles such as rectangles and squares, the interior angles must be 90°.
In rhombuses, isosceles trapezoids, and parallelograms, the interior angles will sort into two pairs of equal angles.
In a trapezium, the interior angles will all be unequal. A shape with different measures for each interior angle could also be called an irregular quadrilateral.
A quadrangle with one right angle could be a trapezium or irregular kite.
Quadrangle sides
Every quadrangle has four sides that connect four vertices on the flat surface (the plane), and every point connects to two other points to make the shape.
The length of the sides tells you a lot about each quadrangle:
If no two sides are the same length, you have a trapezium.
If two sides of equal length, you have an isosceles trapezoid.
If two pairs of opposite sides are equal, you have a parallelogram, rectangle, or square.
If two pairs of adjacent sides are equal, you have a kite or dart.
The sides must be comprised of straight-line segments. If the sides are curved, it is not a quadrangle.
Quadrangles examples
Nine different shapes are examples of quadrangles:
Trapezium
Trapezoid
Isosceles trapezoid
Kite
Dart
Parallelogram
Rectangle
Rhombus
Square
Types of quadrangles
Quadrangles can be concave or convex. A concave quadrangle has one interior angle greater than 180°.
A convex quadrangle has no interior angle greater than 180°, so all of its sides seem to push out. All the other shapes we have seen, except for the dart, have been convex quadrangles.
A special case of quadrangles can be made if we let two sides intersect each other. These are complex quadrangles, more accurately called complex quadrilaterals.
These geometric shapes are self-intersecting. They have two acute (less than 90°) interior angles and two interior angles that are greater than 180° and appear to be outside the shape!
Quadrangles (or quadrilaterals) can fit together in interesting ways. For examples, a square is really several things:
It’s a trapezium because it has four straight sides closing in an area
It’s a kite because it has two pairs of congruent sides that are adjacent
It’s a rhombus because its four sides are equal lengths
It’s a parallelogram because it has one pair of opposite sides parallel
The most accurate description, though, is that it is a square.
Quadrangle quiz
Is a trapezoid a quadrangle?
How many sides does a quadrangle have?
Is a square a quadrangle?
Name at least one quadrangle that is not a parallelogram.
We know all the angles. Don’t look at the answers until you try all the questions!
A trapezoid is a quadrangle. It is a flat shape with four straight sides that close in a space.
A quadrangle has four sides.
A square is a quadrangle.
Quadrangles that are not parallelograms are kites, darts, trapezoids, complex quadrilaterals, and trapeziums.
The quadrangle shape from the image is the trapezoid because it has four straight sides closing in an area. The others are not quadrangles because they do not close in a space, have too few sides, have too many sides, or have curved sides.