Quadrilaterals: Classifying, Perimeter, and Area |

**Quadrilaterals: Four-Sided Polygons**

**Definition:** a polygon is a dead parrot!

In a joke 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 **pent**agon has **5** angles and 5 sides, and a **deca**gon has **10**, etc.

The **stop sign** on the corner is a **regular** **oct**agon -- 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

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

.

**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

Actually

A

A

The angles are

Actually,

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**.

.

**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,

In the same way, it should be obvious that

the perimeter formula for a

and for a

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**.

.

**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

Or, we could realize that 9 inches = ¾ of a foot,

so the

.

**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*".

.

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 |

.

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.

.

**Practice Exercises**

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

trapezoid | a) //gm with 4 equal sides |

rectangle | b) quadrilateral with opposite sides parallel |

square | c) 2 sides parallel, 2 sides not parallel |

360° | d) Area formula for a rectangle |

rhombus | e) Perimeter formula for a square |

l × w |
f) rectangle with 4 equal sides |

4s |
g) Area formula for a trapezium |

parallelogram | h) //gm with 4 right angles |

½h (a + b ) |
i) sum of the angles in a quadrilateral |

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.

trapezoid | c) 2 sides parallel, 2 sides not parallel |

rectangle | h) //gm with 4 right angles |

square | f) rectangle with 4 equal sides |

360° | i) sum of the angles in a quadrilateral |

rhombus | a) //gm with 4 equal sides |

l × w |
d) Area formula for a rectangle |

4s |
e) Perimeter formula for a square |

parallelogram | b) quadrilateral with opposite sides parallel |

½h (a + b ) |
g) Area formula for a trapezium |

2)

figure name |
Perimeter |
Area |

a) Parallelogram | P = 2( = 33 inchest + s) |
= 12 × 3.7 = 44.4 in²A = b × h |

b) Rectangle | P = 2( = 8.48 feetl + w) |
= 3.18 × 1.06 = 3.37 ft²A = l × w |

c) Square | P = 428 cms = |
= 49 cm²A = s² |

d) Trapezium | P = sum of sides = 29.3 yds |
= ½(4)(17.5) = 35 yd ²A = ½h ( a + b ) |

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

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

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

find the dimensions.

( *Plane Geometry MathRoom Index* )

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