Quadrilaterals

Quadrilaterals

Quadrilateral just means "four sides"
(quad means four, lateral means side).

A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

Try it Yourself

(Also see this on Interactive Quadrilaterals)

Properties

Quadrilateral Angles

Try drawing a quadrilateral, and measure the angles. They should add to 360°

Types of Quadrilaterals

There are special types of quadrilateral:

Types of Quadrilateral

Some types are also included in the definition of other types! For example a square, rhombus and rectangle are also parallelograms. See below for more details.

Let us look at each type in turn:

The Rectangle

Rectangle
 
right angle key
means "right angle"
 equal angle keyandequal angle key 2
show equal sides
 

A rectangle is a four-sided shape where every angle is a right angle (90°).

Also opposite sides are parallel and of equal length.

The Rhombus

Rhombus

A rhombus is a four-sided shape where all sides have equal length.

Also opposite sides are parallel and opposite angles are equal.

Another interesting thing is that the diagonals (dashed lines in second figure) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.

A rhombus is sometimes called a rhomb or a diamond.

The Square

Square
 
right angle key
means "right angle"
equal key
show equal sides
 

A square has equal sides and every angle is a right angle (90°)

Also opposite sides are parallel.

A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

The Parallelogram

Parallelogram

A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles "a" are the same, and angles "b" are the same).

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

Example:

Square

A parallelogram with:

is a square!

 

The Trapezoid (UK: Trapezium)

Trapezoid (or Trapezium)
Trapezoid
Isosceles Trapezoid

A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.

And a trapezium (called a trapezoid in the UK) is a quadrilateral with NO parallel sides:

  Trapezoid Trapezium
In the US: a pair of parallel sides NO parallel sides
 
In the UK: NO parallel sides a pair of parallel sides
(the US and UK definitions are swapped over!)

(Note: when the two sides joining parallel sides are equal in length and both angles coming from a parallel side are also equal we call it an Isosceles trapezoid, as shown above.)

The Kite

The Kite

Hey, it looks like a kite (usually).

It has two pairs of sides.

Each pair is made up of adjacent sides (they meet) that are equal in length.

The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.

 

... and that's it for the special quadrilaterals.

 

Irregular Quadrilaterals

The only regular (all sides equal and all angles equal) quadrilateral is a square. So all other quadrilaterals are irregular.

 

The "Family Tree" Chart

Quadrilateral definitions are inclusive.

Example: a square is also a rectangle.

So we include a square in the definition of a rectangle.

(We don't say "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")

This may seem odd, as in daily life we think of a square as not being a rectangle ... but in mathematics it is.

Using the chart below we can answer such questions as:

Quadrilateral Classification

Complex Quadrilaterals

Oh Yes! when two sides cross over, we call it a "Complex" or "Self-Intersecting" quadrilateral, like these:

Complex Quadrilaterals

They still have 4 sides, but two sides cross over.

Polygon

A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on.

Play with Them

Now that you know the different types, you can play with the Interactive Quadrilaterals.

Other Names

A quadrilateral can sometimes be called: