Quadrilaterals: Four-Sided Polygons

Definition: a polygon is a dead parrot!

In a Monty Python's skit maybe, but in Geometry,

a polygon is a closed figure with three or more straight sides.

A polygon of n sides has n interior and n exterior angles.

In a Regular polygon all sides, all interior and all exterior angles are equal.

We name polygons by the number of sides and/or the number of angles in them.
A triangle has 3 angles which it's name declares.
a quadrilateral has 4 sides -- quad is means 4, and lateral means side

A lateral pass in football is a pass to the side instead of forward.

a pentagon has 5 angles and 5 sides, and a decagon has 10, etc.
The stop sign on the corner is a regular octagon -- with 8 equal sides. The tiled walls and floors in our buildings and homes are made of regular polygons formed into patterns. Our yards, buildings, furniture and clothing designs are all shaped like various polygons so they are found all around us.

This lesson covers the 4 sided polygons called quadrilaterals.

Classifying Quadrilaterals

We classify quadrilaterals by certain properties of their sides and their angles. The classification property of the sides is parallelism -- how many (if any) pairs of sides are parallel.
The angle property we consider is whether they are 90° or not.

The sum of the interior angles of any quadrilateral is 360°.

A kite, lozenge, or just quadrilateral (like the yellow one above) has 0 pairs of parallel sides and could have a right angle or not. It always has 4 sides.
In common terms, we often call it a diamond.

A parallelogram is a quadrilateral with both pairs
of opposite sides parallel to each other.

Though the definition doesn't include it, we can see that the opposite sides are also equal.
The opposite internal angles are equal as well.
And the diagonals bisect each other -- they meet at their midpoints.

Notice how we use symbols in the diagram to show which sides are equal and parallel.
We use > or >> to indicate parallelism, and | or // to indicate equality.
Equal angles are indicated either by arcs like in the upper image, or by
lower case letters such as x and o shown in the lower image.

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3 Types of Parallelogram

The term parallelogram applies to any quadrilateral that has parallel opposite sides. There are
3 sub-classes of parallelogram with special names we use all the time. They are:

RECTANGLE -- SQUARE -- RHOMBUS

A rectangle is a parallelogram that has 4 right angles.
Actually, rectangle means "right" angle.
A square is a rectangle with 4 equal sides.
A rhombus is a parallelogram with 4 sides of equal length.
The angles are not necessarily right angles. It's like a mangled square.
Actually, a square is a rhombus with 90° angles.

There is one more quadrilateral to classify and that's the Trapezoid or Trapezium --
which has 2 sides parallel and 2 sides not. When the 2 "not parallel" sides are equal,
it is called an isosceles trapezoid or trapezium.

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Getting Around Quadrilaterals: Perimeter

The Perimeter of a figure is the distance or measure around the outside, so if we have a quadrilateral with 4 unequal sides, we just add their lengths to find the Perimeter. The blue trapezoid above is such a case.

Since a parallelogram has 2 pairs of equal sides, we can find a formula for the Perimeter.
If the top and bottom measure " t " units and the sides measure " s " units,
the perimeter formula will be P = 2 t + 2 s, or, more efficiently, P = 2 ( t + s ).
In the same way, it should be obvious that
the perimeter formula for a rectangle with length = l, width = w is P = 2 l + 2 w = 2 ( l + w )
and for a square or a rhombus it is P = 4 s, where the length of a side = s.

The length and width of a rectangle are also called it's base and height.

Examples:

1) Find the Perimeter of:

a) A parallelogram (//gm) with top/bottom = 7 inches and sides = 9 inches
Perimeter = 2 (7 + 9) = 2 (16) = 32 inches.

b) A rectangle with length (l) = 14 ft. and width (w) = 10 ft.
Perimeter = 2 (14 + 10) = 2 (24) = 48 feet.

c) A Square with side (s) = 12 yards.
Perimeter = 4 (12) = 48 yards.

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BEWARE!! of mixed units!! If one measure is inches, the other is feet, we have to convert in order to have units of the same size to add and multiply. Answers must include units!!

Example: Find the Perimeter of a parallelogram (//gm) with top/bottom = 2 feet, side = 9 inches.
Solution: We could change feet to inches: 2 feet = 2 × 12 = 24 inches
so the Perimeter = 2 (24 + 9) = 66 inches.
Or, we could realize that 9 inches = ¾ of a foot,
so the Perimeter = 2 (2 + ¾) = 4 + 1½ = 5½ feet which also equals 66 inches.

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Covering the Surface of Quadrilaterals: Area

The things we use -- like paint, wallpaper and carpets -- to cover surfaces such as walls and floors are always measured and priced in square units. The information on every paint can label includes the number of square feet or square meters of wall space, the paint will cover.

The measure of the surface covered by a closed figure is called AREA.
We measure area in SQUARE UNITS -- square inches, square feet, square miles --
because, to measure the area of a figure, we "square it off" and then count the squares.

We see, that to completely cover the pink square ( 5 × 5 ), with grey squares ( 1 × 1 ),
we would use 5 rows of 5 squares each -- so we'd need 25 squares in all.
Area of a square that is 5 by 5 = 25 units².
Area of a square that is 9 inches per side = 81 in² (square inches).
That's exactly why we call " a " to the 2nd power "a-squared".

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Now let's discover how to "square off" a quadrilateral with no right angles.

Let's look at a parallelogram and a trapezium to find the formulas for their Areas.

Area Formulas for Quadrilaterals
quadrilateral name	Area Formula	Comments
Square	A = s²	all sides = s
Rectangle	A = l × w	length = l, width = w
Parallelogram	A = b × h	base = b, perpendicular height = h
Trapezium	A = ½ h ( a + b )	bases = a and b, perpendicular height = h

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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

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Practice Exercises

1) Match the words in the left column with the descriptions on the right.

2) For each image in the diagram, write the name of the figure, find the perimeter, and area.
(be sure to write the formula used for both perimeter and area)

3) Draw a diagram, then solve each of these questions:
a) The Area of a rectangle = 36 in². If the length l = 9 in, find the width w.
b) The Perimeter of a Square = 44 in. Find the length of the side s and the Area.
c) The Perimeter of a parallelogram is 96 feet. If the base is twice as long as the side,
find the dimensions.

Solutions

1) Match the words in the left column with the descriptions on the right.

2)

figure name	Perimeter	Area
a) Parallelogram	P = 2(t + s) = 33 inches	A = b × h = 12 × 3.7 = 44.4 in²
b) Rectangle	P = 2(l + w) = 8.48 feet	A = l × w = 3.18 × 1.06 = 3.37 ft²
c) Square	P = 4s = 28 cm	A = s² = 49 cm²
d) Trapezium	P = sum of sides = 29.3 yds	A = ½h ( a + b ) = ½(4)(17.5) = 35 yd ²

3) a) Area = 36 in². l = 9 in, w = 36 ÷ 9 = 4 inches.

b) Perimeter = 44 in. 4s = 44 so s = 11 in. Area = 11² = 121 in²

c) The Perimeter of a parallelogram is 96 feet. If the base is twice as long as the side,
find the dimensions.

( Primary MathRoom Index )