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Quadrilaterals

Quadrilaterals

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

Any four-sided shape is a Quadrilateral.

But the sides have to be straight, and it has to be 2-dimensional.

Try for Yourself

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Properties

  • Four sides (or edges)
  • Four vertices (or corners).
  • The interior angles add up to 360 degrees:

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

If it isn't one of those it is an irregular 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
 
means "right angle"
and
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) of a rhombus bisect each other at right angles.

The Square

Square
 
means "right angle"
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

Opposite sides are parallel and equal in length, and opposite angles are equal (angles "a" are the same, and angles "b" are the same)

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

Example: if a parallelogram has all sides equal and angles "a" and "b" are right angles, then it is also a square.

 

The Trapezoid (UK: Trapezium)

Trapezoid (or Trapezium)
Trapezoid
Isosceles Trapezoid

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

It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown.

A trapezoid is not a parallelogram because only one pair of sides is parallel.

Language Note: In the US a "trapezium" is a quadrilateral with NO parallel sides!

The Kite

The Kite

Hey, it looks like a kite. It has two pairs of sides. Each pair is made up of adjacent sides 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; if it doesn't fit one of those it is an Irregular Quadrilateral:

Irregular Quadrilaterals

Irregular Quadrilaterals

Any quadrilateral that doesn't match one of the previous types.

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 "A rectangle has all 90° angles, except if it is a square")

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

Using the chart below you can answer such questions as:

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

Quadrilateral Classification

Complex Quadrilaterals

Oh Yes! when two sides cross over, you 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:

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