Quadrilaterals

A quadrilateral is a closed figure with four straight sides. You can make a quadrilateral by taking (or imagining) anything straight and thin you might have handy: pens, toothpicks, chopsticks, etc. A square is one type of a special quadrilateral.

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.

Properties

A quadrilateral has:

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

Exercise

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.

Let us look at each type in turn:

The 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

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

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/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.https://googleads.g.doubleclick.net/pagead/ads?guci=2.2.0.0.2.2.0.0&client=ca-pub-1661929042807246&output=html&h=280&slotname=9757662299&adk=416771370&adf=2768648106&pi=t.ma~as.9757662299&w=700&fwrn=4&fwrnh=100&lmt=1604309646&rafmt=1&psa=1&format=700×280&url=http%3A%2F%2Fstoplearn.com%2Fcourses%2Fsecondary-school%2Fjss2-second-term-basic-technology-junior-secondary-school%2Flessons%2Fquadrilaterals%2F&flash=0&fwr=0&fwrattr=true&rpe=1&resp_fmts=3&wgl=1&adsid=ChAIgPn-_AUQkJWp-eO5jtJNEkgAMWjS4M82Y6pcyVlsvJo4rLAchUMSuzz8-GT7dd5X676fMlzH2dtq4UhjTTcYNXyzJMfGneUrUBXwbPUvgQNoXt8Y2f2dD9A&tt_state=W3siaXNzdWVyT3JpZ2luIjoiaHR0cHM6Ly9hZHNlcnZpY2UuZ29vZ2xlLmNvbSIsInN0YXRlIjowfSx7Imlzc3Vlck9yaWdpbiI6Imh0dHBzOi8vYXR0ZXN0YXRpb24uYW5kcm9pZC5jb20iLCJzdGF0ZSI6MH1d&dt=1604309619554&bpp=10&bdt=159&idt=112&shv=r20201029&cbv=r20190131&ptt=9&saldr=aa&abxe=1&cookie=ID%3D3b5c0a4b352ccd7b-2255262b41a60032%3AT%3D1602230088%3ART%3D1602230088%3AS%3DALNI_MaWUG7yiaaHthyGgrREyU5I0uWdRw&prev_fmts=0x0%2C1200x90_0ads_al&nras=1&correlator=7818172716258&frm=20&pv=1&ga_vid=1300243797.1604309620&ga_sid=1604309620&ga_hid=176727478&ga_fc=0&iag=0&icsg=599886135210&dssz=59&mdo=0&mso=0&u_tz=60&u_his=20&u_java=0&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=24&u_nplug=3&u_nmime=4&adx=609&ady=658&biw=1518&bih=667&scr_x=0&scr_y=0&eid=21066922&oid=3&psts=AGkb-H-ugxHRqvUGENiri2BIZnLdH5w6PJpzzPVfFOWi4qPPWfUfo2puFA&pvsid=4371205875330035&pem=804&ref=http%3A%2F%2Fstoplearn.com%2Fcourses%2Fsecondary-school%2Fjss2-second-term-basic-technology-junior-secondary-school%2F&rx=0&eae=0&fc=896&brdim=0%2C0%2C0%2C0%2C1366%2C0%2C1366%2C728%2C1517%2C666&vis=1&rsz=%7C%7CaoeE%7C&abl=CA&pfx=0&fu=8320&bc=31&jar=2020-10-31-22&ifi=1&uci=a!1&fsb=1&xpc=Tw6NbWRwPA&p=https%3A//stoplearn.com&dtd=26579

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.

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.

ASSESSMENT

1. What is the name of this quadrilateral?
   (a) kite
   (b) square
   (c) rhombus
   (d) trapezium
2. What is the name of this quadrilateral?(a) trapezoid or trapezium
   (b) kite
   (c) parallelogram
   (d) rhombus
3. What is the name of this quadrilateral?(a) kite
   (b) parallelogram
   (c) rhombus
   (d) trapezoid or trapezium
4. What is the name of this quadrilateral?(a) parallelogram
   (b) rectangle
   (c) trapezoid or trapezium
   (d) kite
5. One of these statements is not true of kites
   (a) it has two pairs of sides
   (b) it has three pairs of sides
   (c) the angles where the two pairs meet are equal
   (d) one of the diagonals bisects (cuts equally in half) the other