# 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

A quadrilateral has:

- four sides (edges)
- four vertices (corners)
- interior angles that add to
**360 degrees**:

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

## Types of Quadrilaterals

There are special 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

means "right angle" |
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and |
show equal sides |
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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

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

means "right angle" |
|||

show equal sides |
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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

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:

A **parallelogram** with:

- all sides equal and
- angles "a" and "b" as right angles

is a **square**!

## The Trapezoid (UK: 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!) |

An **Isosceles** trapezoid, as shown above, has left and right sides of equal length that join to the base at equal angles.

## The Kite

Hey, it looks like a kite (usually).

It has **two pairs** of sides:

Each pair is made of two equal-length sides that join up.

Also:

- the angles where the two pairs meet are equal.
- the diagonals, shown as dashed lines above, meet at a right angle.
- one of the diagonals
*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:

- Is a Square a type of Rectangle? (Yes)
- Is a Rectangle a type of Kite? (No)

## Complex Quadrilaterals

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

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:

- a
**Quadrangle**("*four angles*"), so it sounds like "triangle" - a
**Tetragon**("*four*and*polygon*"), so it sounds like "pentagon", "hexagon", etc.