Area and perimeter.docx

8/22/2019 Area and perimeter.docx

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Areaandperimeterare two calculations performed on manygeometric shapes. Perimeter is a measure of

distance around a shape; for example, someone might want to figure out the perimeter around their

garden before buying material to make a fence so that they know how much material to buy. Area is a

measure of the amount of surface something covers; for example, someone might want to know how

much space their garden takes up. Area and perimeter are often grouped together because one can be

used to help you figure out the other. For example, if you know the perimeter of a square, you can easily

figure out the area, and vice versa.

PerimeterPerimeter simply measures the distance around an area. It can be measured in inches, feet, yards, miles,

centimeters, meters, kilometers, and so on (any standard distance measurement). You can measure the

perimeter of nearly any shape, you just add together the measure of each of its sides. Much of the ability

to figure out perimeter lies in remembering the properties of certain shapes. Well go through several

examples.

Perimeter of a Square

Taking the following square with side length 6 inches, calculate the perimeter.

In order to calculate perimeter, you need to add together the lengths of all four sides of the square. You

are given the length of one side. Remember, all sides of a square are equal, so really you already have

the measures of each side.

Then, you add them together, so 6 + 6 + 6 + 6 = 24 inches. Thus, 24 inches is your final answer.

Perimeter of a Rectangle

Taking the following rectangle with length 8 inches and width 4 inches, calculate the perimeter.
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In order to calculate perimeter, you need to add together the lengths of all four sides of the rectangle.

You are given the length of one side and the width of one side. Remember, opposite sides of a rectangle

are equal, so really you already have the measures of each side.

Then, you add them together, so 8 + 8 + 4 + 4 = 24 inches. Thus, 24 inches is your final answer.

Perimeter of a Polygon

The perimeter of a polygon is calculated using the same method of adding together each side. Remember

that if all the sides are equal, you only need to know one side of the polygon. If the sides are unequal,

however, you do need to know the length of each different side. Taking the following pentagon with side

length 7, calculate the perimeter.

A pentagon has five sides, and all of these sides are equal, therefore you can perform the following

calculation:

7 + 7 + 7 + 7 + 7 = 35

Example 1

Michelle was planting a garden. She wanted her garden to be fenced in, so she went to the hardware

store to buy fencing material. The salesperson asked Michelle how big her garden would be. She thought

about it, and then replied that her garden would be 4 feet wide, and that it would be 2 feet longer (inlength) than it is wide. Answer the following questions:

1. What shape is Michelles garden?

2. How long is Michelles garden?

3. What is the perimeter of Michelles garden?


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4. Draw and label Michelles garden.

Solution

Once you've worked out the answers, click "Next Step" to show the answer to each answer to each

question!

1. Michelles garden is a rectangle. We know this because it talks about length and width (4-sided shapes

have these) and we can conclude it is not a square, because the length and width are different, therefore

all four sides are not equal.

2. The problem stated that Michelles garden is two feet longer than it is wide. We know Michelles

garden is 4 feet wide, so we know that we have to add 2 to that number, resulting in 6. Thus, Michelles

garden is 6 feet long.

3. In order to find the perimeter of Michelles garden, we have to add together all four sides. We know

that two sides are 4 feet long, and the other two sides are 6 feet long. Therefore, we can solve the

addition problem: 4 + 4 + 6 + 6 = 20 feet. Our answer is that the perimeter of her garden is 20 feet.

This means that, when Michelle buys the material to build her fence, shell need 20 feet of material in

order for the fence to be complete.

4.

Example 2

Andrew is going to build a box to hold his hats. He decides that each side should be 5 inches long. He

also decides to make this box in the shape of a regular hexagon. Answer the following questions:

1. How many sides does Andrews box have?

2. Are all sides the same length? How do you know?

3. What is the perimeter of to Andrews box?


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Solution

Once you've worked out the answers, click "Next Step" to show the answer to each answer to each

question!

1. We know that Andrews box is in the shape of a hexagon, and a hexagon has 6 sides. Therefore,

Andrews box also has 6 sides.

2. All sides of Andrews box are the same length. We know this because the problem stated that we have

a regular hexagon, and we know that regular means all sides are the same.

3. We can easily calculate the perimeter of Andrews lid to the box by using the following addition

problem: 5 + 5 + 5 + 5 + 5 + 5 = 30 inches.

Area

Area is the measure of the amount of surface covered by something. Area formulas for differentshapesare sometimes different, but for the most part, area is calculated by multiplying length times width. This

is used when calculating area of squares and rectangles. Once you have the number answer to the

problem, you need to figure out the units. When calculating area, you will take the units given in the

problem (feet, yards, etc) and square them, so your unit measure would be in square feet (ft.2) (or

whatever measure they gave you).

Area Example 1

Lets try an example. Nancy has a vegetable garden that is 6 feet long and 4 feet wide. It looks like this:
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Nancy wants to cover the ground with fresh dirt. How many square feet of dirt would she need?

We know that an answer in square feet would require us to calculate the area. In order to calculate the

area of a rectangle, we multiply the length times the width. So, we have 6 x 4, which is 24. Therefore,

the area (and amount of dirt Nancy would need) is 24 square feet.

Area Example 2

Lets try that one more time. Zachary has a wall that he would like to paint. The wall is 10 feet wide and

16 feet long. It looks like this:

Using Area and Perimeter Together

Sometimes, you will be given either the area or the perimeter in a problem and you will be asked to

calculate the value you are not given. For example, you may be given the perimeter and be asked to

calculate area; or, you may be given the area and be asked to calculate the perimeter. Lets go through a

few examples of what this would look like:

Area and Perimeter Example 1

Valery has a large, square room that she wants to have carpeted. She knows that the perimeter of the

room is 100 feet, but the carpet company wants to know the area. She knows that she can use the

perimeter to calculate the area.

What is the area of her room?

We know that all four sides of a square are equal. Therefore, in order to find the length of each side, we

would divide the perimeter by 4. We would do this because we know a square has four sides, and they

are each the same length and we want the division to be equal. So, we do our division100 divided by

4and get 25 as our answer. 25 is the length of each side of the room. Now, we just have to figure out

the area. We know that the area of a square is length times width, and since all sides of a square are the

same, we would multiply 25 x 25, which is 625. Thus, she would be carpeting 625 square feet.


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Area and Perimeter Example 2

Now lets see how we would work with area to figure out perimeter. Lets say that John has a square

sandbox with an area of 100 square feet. He wants to put a short fence around his sandbox, but in order

to figure out how much fence material he should buy, he needs to know the perimeter. He knows that he

can figure out the perimeter by using the area.

What is the perimeter of his sandbox?

We know that the area of a square is length times width. In the case of squares, these two numbers are

the same. Therefore, we need to think, what number times itself gives us 100? We know that 10 x 10 =

100, so we know that 10 is the length of one side of the sandbox. Now, we just need to find the

perimeter. We know that perimeter is calculated by adding together the lengths of all the sides.

Therefore, we have 10 + 10 + 10 + 10 = 40 (or, 10 x 4 = 40), so we know that our perimeter is 40 ft.

John would need to buy 40 feet of fencing material to make it all the way around his garden.

Calculating Area and Perimeter Using Algebraic Equations

So far, we have been calculating area and perimeter after having been given the length and the width of

a square or rectangle. Sometimes, however, you will be given the total perimeter, and a ratio of one side

to the other, and be expected to set up an algebraic equation (using variables) in order to solve the

problem. Well show you how to set this up so that you can be successful in solving these types of

problems.

Eleanor has a room that is not square. The length of the room is five feet more than the width of the

room. The total perimeter of the room is 50 ft. Eleanor wants to tile the floor of the room. How manysquare feet (ft 2) will she be tiling?

In this problem, we will be calculating area, but first were going to use the perimeter to figure out the

length and width of the room.

First, we have to assign variables to each side of the rectangle. X is the most often used variable, but you

can pick any letter of the alphabet that youd like to use. For now, well just keep things simple and use

x. To assign a variable to a side, you first need to figure out which side they give you the least

information about. In this problem, it says the length is five feet longer than the width. That means that

you have no information about the width, but you do have information about the length based on the

width. Therefore, youre going to call the width (the side with the least information) x. Now, the width =

x, and x simply stands for a number you dont know yet. Now, you can assign a variable to the length.

We cant call the length x, because we already named the width x, and we know that these two

measurements are not equal. However, the problem said that the length is five feet longer than the

width. Therefore, whatever the width (x) is, we need to add 5 to get the length. So, were going to call

the length x + 5.


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Now that weve named each side, we can say that width =x, and length = x + 5. Heres a picture of

what this would look like:

Next, we need to set up an equation using these variables and the perimeter in order to figure out the

length of each side. Remember, when calculating perimeter you add all four sides together. Our equation

is going to look the same way, just with xs instead of numbers. So, our equation looks like this:

x + x + x + 5 + x + 5 = 50

Now, we need to make this look more like an equation we can solve. Our first step is to combine like

terms, which simply means to add all the xs together, and then add the whole numbers together (for

more help on this, seeCombining Like Terms).

Once we combine like terms, our equation looks like this:

4x + 10 = 50

Next, we follow the steps for solving equations. (For additional help with this, seeSolving Equations). We

subtract 10 from each side of the equation, which leaves us with the following:

4x = 40

Now, we have to get x by itself, which means getting rid of the 4. In order to do this, we need to perform

the opposite operation of whats in the equation. So, since 4x means multiplication, we need to divide by

4 to get x alone. But remember, what we do to one side, we have to do to the other side. After dividing

each side by 4, we get:

x = 10

Next, we have to interpret what this means. We look back and recall that we named the width x, so the

width is 10. Now, we need to figure out the length. We named the length x + 5, so that means we have
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to substitute 10 in for x, and complete the addition. Therefore, we have 10 + 5, which gives us 15. So,

our length is 15.

Now, we need to look back and remember that the problem asked us to calculate the area of the floor

that Eleanor will be tiling. We know that in order to calculate area, we need to multiply the length times

the width. We now have both the length and the width, so we simply set up a multiplication problem, like

this: 10 x 15 = ? We multiply the two numbers together, and get 150.

Thus, your final answer is 150 ft 2.

Area and Perimeter Practice Problems

Now, well give you several practice problems so that you can try calculating area and perimeter on your

own.

1. Leah has a flower garden that is 4 meters long and 2 meters wide. Leah would like to put bricks

around the garden, but she needs to know the perimeter of the garden before she buys the bricks.

What is the perimeter of Leahs garden (in meters)?

{12| 12 m| 12 meters| 12 meter}

2. David has a rug that is square, and the length of one side is 5 feet. He has an open floor space in his

living room that is 36 square feet.

Would the rug fit perfectly, be too big, or be too small for the space he has? (Answer Choices: fitperfectly, too big, too small)

too small

3. Debbie has pool in her back yard that has a perimeter of 64 feet. The length of the pool is 2 feet

longer than the width. Debbie wants to buy a cover for the pool, and needs to know how many square

feet she needs to cover.

How many square feet (ft2) is Debbies pool? (hint: if you can, set up an algebraic equation to solve thisproblem!)

{255|255 sq ft|255 square feet|255 ft2|255 ft|255 ft2}

4. Hector is planting a square garden in front of his house. He wants to plant carrots in the garden. He

knows he can plant the carrots one foot apart. He has six feet across his yard (length) and he can plant

carrots four feet deep (width).

How many square feet (ft2) does he have to plant carrots?


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{24| 24 ft sq|24 sq ft|24 ft}

5. Amanda is building a house, and shes trying to calculate the area of her bedroom. She knows that the

living room is 22 feet long and 20 feet wide. She was told that her bedroom should be half of the area of

the living room.

What will the area of her bedroom be?


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Perimeter, Area and Volume

This topic makes use of formulae for perimeters, areas and volumes of shapes and solids. If you

need help with working with formulae see the topic Formulae and Algebra in the menu to the left

of the screen before reviewing this topic.

Note - this topic contains specialised formatting and symbols. If you copy parts of it to the

Scratch Pad this formatting may be lost. You can print the topic without losing the formatting and

symbols. There are 10 pages.

You can scroll down to read all the help in this topic or click on one of the links below to go straight

to a specific area.

Click on one of these links to go to help on working withperimeter,areaandvolume, alternatively

you may want to check a particular formula, click on any of the following to go straight to it:

Calculating

Circumference of a circle ,Circumference of an ellipse Area ofsquare/rectangle,triangle,parallelogram,trapezium,any other polygon,circle Volume ofcube/cuboid,pyramid,cone,cylinder,sphere

Perimeter

What is perimeter? The perimeter is the distance or length around the

outside of a shape.

It is calculated by adding the lengths of all the sides

together, ie in diagram (left):

Perimeter = 2 + 6+ 4+ 6= 18 cm

As perimeter is a length or distance it is

measured in units of length, eg mm, cm, etc.
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It may help if you think of it as how far you

would have to walk, to go all the way round

the edges of the shape. Also if you are

working out the perimeter of a complicated

shape, it may help to mark your starting point

(see left). This way you can be sure that you

add all the sides once and once only.

What is circumference?The perimeter of a circle, and also an ellipse (or

oval), is called its circumference.

The perimeter of shapes with straight edges can be

measured using a ruler and adding the sides together

as explained above. However, it is not possible to

accurately measure a curve using a straight ruler. A

different method is used to work out the

circumferences of circles and ellipses.

Calculating the circumference of a circle. The circumference of a circle is calculated using the

following formula:

Circumference = 2 rwhere:

r = the radius of the circle.This is sometimes written as:

Circumference = dwhere:

d = the diameter of the circle.NB The radius of the circle is the distance


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from the centre to the edge (see diagram

left). The diameter of the circle is the length

of a line going from one edge to another,

passing through the centre. This is the same

as twice the radius.

What is ?(pi) is the number that represents the ratio

between the radius and diameter of a circle to its

circumference and area. It has an endless number of

decimal places. Here it is shown rounded to 8 decimal

places

3.14159265.

If you have a scientific calculator you will have a ' '

button on it.

Example 1

Calculate the circumference of a circle

whose radius is3 cm.Using the formula:

Circumference = 2 rIn the example, r = 3.

So:

Circumference= 2 x 3.14159265 x 3

= 18.8495559= 18.8496 cm (4 dp)

For more help with rounding decimals and decimal


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places see the sub-topic Decimals in the menu to the

left of the screen.

Circumference of an ellipse. The circumference of an ellipse is difficult tocalculate exactly. If you require the formula for the

best approximation of this, look in a mathematics

study dictionary. One is recommended in the

'Resources You Can Use' for this topic.Area

What is area? The area is the surface space contained within the

edges of a 2-D shape (see coloured area of shape

left).

Area is measured as the number of squares of a

particular unit, eg mm2, cm2, m2 etc.


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Calculating the area of a square or

rectangle.The shape, left, has been divided into squares. Each

square is 1 cm2 (1 cm x 1 cm).

The area of the rectangle is calculated using the

formula:

Area = l x bwhere:

l = lengthb = breadth.Here, l = 5 cm and b = 2 cm.

So:

Area= 5 x 2= 10 cm2

You can check the answer by counting how

many 1 cm2 squares there are within the

shape.

The same formula is used for the area of a

square. As all the sides of a square are the

same length, this will be a number multiplied

by itself.


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Example 2

What is the area of a square with sides6

cm?Using the formula Area = l x b

where:

l = length

b = breadth.

Here, l and b both equal 6 cm.

So:

Area= 6 x 6= 36 cm2

This is the same as 62 (this is said as 'six squared'

or 'six to the power two'). For help with powers see

the sub topic Powers and Roots in the menu to the

left of the screen.

Calculating the area of a triangle.The area of a triangle is calculated using the

formula:

Area = x b x h

where:

b = the length of the base of triangle

h = the perpendicular height of trianglesee the

diagram left.

Example 3

What is the area of a triangle with a base

length of5 cm and perpendicular height of

3 cm?

Using the formula:

Area = x b x hIn the example, b = 5 and h = 3.

So:

Area= x 5 x 3


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= 7.5 cm2

Calculating the area of a parallelogram.A parallelogram is a four-sided shape where both

pairs of opposite sides are parallel. This means that

they are always the same distance apart and can

never meet however far they are extended.

The area of a parallelogram is calculated using the

formula:

Area = b x hwhere:

b = the length of the base of the

parallelogram

h = the perpendicular height of parallelogramsee the diagram left.

Example 4

What is the area of a parallelogram with

base length6 cm and perpendicular height

2 cm?Using the formula Area = b x h

In the example, b = 6 and h = 2.

So:

Area= 6 x 2= 12 cm2

Calculating the area of a trapezium.A trapezium is a four-sided shape with one pair of

parallel sides.

The area of a trapezium is calculated using the


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formula:

Area=(b + p) x h2

where:

b = the length of the base of trapezium p = the length of the parallel edge

h = the perpendicular height of trapezium. NB. The base of the trapezium need not be the

bottom of the shape, but it should be one of the

parallel sides. Therefore when calculating the area,

(b + p) means add together the lengths of the two

parallel sides.

Example 5

What is the area of a trapezium with

parallel sides of length8 cm and6 cm?

Its perpendicular height is3 cm.

Using the formula:Area= (b + p) x h

2

In the example, b = 8, p = 6 and h = 3.

So:

Area =(8 + 6) x 32

= 14 x 3


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2= 21 cm2

Calculating the area of other polygons. A polygon is a 2-D shape enclosed by three or moresides. Triangles, squares, rectangles etc are all

polygons. Shapes with more than four sides, eg.

pentagon (5 sides), hexagon (6 sides), octagon (8

sides) etc. are also polygons.

Each of these have special rules for calculating their

area, if you need to calculate one of these look in a

mathematics study dictionary. One is recommended

in the 'Resources You Can Use' for this topic.

Calculating the area of a circle. The area of a circle is calculated using the following

formula

Area = r2where:

r = the radius of the circle. This is sometimes written as:

Area = 1/4( d2)

where:


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d = the diameter of the circle.is explained above in the section about

calculating the circumference of a circle.

Example 6

What is the area of a circle with a radius

of5cm? Using the formula Area = r2

In the example, r = 5.

So:

Area=3.14159265 x (5)2=3.14159265 x 25=78.5398125=78.54 cm2 (2 dp)

For help with rounding decimals and decimal places

see the sub topic Decimals in the menu to the left

of the screen.

Volume.

What is volume?

The volume, or capacity, of a 3-D shape is how muchspace is contained within the shape. It may help to

think of the 3-D shape (called a solid) as a vessel

that can be filled with a certain volume of liquid.

Volume is measured as the number of cubes of a


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particular unit, eg mm3, cm3, m3 etc.

Calculating the volume of a cube or cuboid. A cube is a solid shape with 6 square faces. A cuboid

is a solid shape with 6 faces that are eitherall

rectangles, ora mixture of rectangles and squares,

see diagram left. Boxes are the most common

examples of cubes and cuboids.

The volume of a cuboid is calculated using the

formula:

Volume = l x b x wwhere:

l = lengthb = breadthw = width.The volume of a cube is calculated in thesame way, however, because all the faces are

square each of these lengths will be the

same.

Example 7

What is the volume of a cube, where each

square face has sides4 cm by4 cm?Using the formula:Volume = l x b x w

where:

l, b, w all equal 4 cm.

So:

Volume= 4 x 4 x 4= 64 cm3


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NB. This is the same as 43 (this is said as 'four

cubed' or 'four to the power three'). For help with

using powers see the sub topic Powers and Roots in

the menu to the left of the screen.

Calculating the volume of a pyramid. A pyramid is a solid. All but one of its faces must be

triangles that meet a point. The base of a pyramid

can be a triangle but does not have to be. The most

common base of a pyramid is a square. However the

base can be any straight-sided shape, ie any polygon.

The number of faces a pyramid has depends on the

number of sides of the base. A square-based pyramid

is shown left.

The volume of a pyramid is calculated using the

formula:

Volume = 1/3Ah

where:

A = area of the baseh = perpendicular height.NB The perpendicular height of a pyramid is the

distance from the apex (top point) straight down to

the base (see h on the diagram left)


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Example 8

A pyramid has a square base with an area

of9 cm2(ie. the lengths of the sides of

the square are each3 cm). It has aperpendicular height of2 cm. What is its

volume?

Using the formula:

Volume = 1/3AhIn the example, A = 9 and h = 2.

So:

Volume= 1/3 x 9 x 2= 6 cm3

Calculating the volume of a cone. The volume of a cone is calculated using the same

formula as for a pyramid:

Volume = 1/3Ahwhere:

A = area of the baseh = perpendicular height.The base of a cone is a circle, so:

Area of the base = r2where:

r = radius of the circle.Therefore the formula can also be written

as:

Volume = 1/3 r2h

where:

r = radius of the circular base h = perpendicular height

is explained above in the section about

calculating the circumference of a circle.


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Example 9

What is the volume of a cone with

perpendicular height of5 cm? The radius

of the circular base is3 cm.Using the formula:

Volume = 1/3 r2h

In the example, r = 3 and h = 5.So:

Volume=1/3 x 3.14159265 x (3)2 x 5= 1/3 x 3.14159265 x 9

x 5= 47.12388975=47.1239 cm3 (2 dp)


places see the subtopic Decimals in the menu to the

left of the screen.

Calculating the volume of a cylinder. The most common cylinders are cans and tubes.

The volume of a cylinder is calculated using the

formula:

Volume = r2h

where

r = radius of one circular end (both ends of a

cylinder will have the same radius)h = height of the cylinder

NB This is the same as multiplying the area of one of

the circular faces by the height of the cylinder.

is explained above in the section about calculating

the circumference of a circle.


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Example 10

What is the volume of a cylinder4 cm

high, whose circular face has a radius of2

cm?

Using the formula:

Volume = r2hIn the example, r = 2 and h = 4.

So:

Volume= 3.14159265 x (2)2 x 4= 3.14159265 x 4 x 4= 50.2654824= 50.27 cm3 (2 dp)

For more help with rounding decimals and

decimal places see the subtopic Decimals in

the menu to the left of the screen.

Calculating the volume of a sphere. A ball is the most common type of sphere seen in

everyday life.

The volume of a sphere is calculated using the

formula:

volume = 4/3 r3

where:

r = the radius of the sphere.This is sometimes written as:

Area = 1/6 d3

where:

d = the diameter of the sphere.NB The radius of a sphere is the distance

from the centre of the sphere to the outer

edge. The diameter of a sphere is the


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distance from one edge to another, going

through the centre. This is the same as twice

the radius.

is explained above in the section aboutcalculating the circumference of a circle.

Example 11

What is the volume of a sphere with a

radius of7 cm?Using the formula Volume = 4/3 r

3

In the example, r = 7.

So:

Volume= 4/3 x 3.14159265 x (7)3Volume= 4/3 x 3.14159265 x 343

= 1436.7550386= 1436.76 cm3 (2 dp)


places see the sub topic Decimals in the menu to the

left of the screen.

Date post:	08-Aug-2018
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