How To Calculate Area

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carpet fitter measuring roomWhat is area?

Area tells us the size of a shape or figure. It tells us the size of squares, rectangles, circles, triangles, other polygons, or any enclosed figure.

In the real world it tells us the size of pieces of paper, computer screens, rooms in houses, baseball fields, towns, cities, countries, and so on. Knowing the area can be very important. Think of getting a new carpet fitted in a room in your home. Knowing the area of the room will help make sure that the carpet you buy is big enough without having too much left over.

Calculating Area

Area is measured in squares (or square units).

How many squares are in this rectangle?

example of rectangle with area of 15 square units

We can count the squares or we can take the length and width and use multiplication. The rectangle above has an area of 15 square units.

 

The area of a rectangle is = length x width

Examples of calculating the area of a rectangle

3 by 2 rectangle

Area = Length x Width

Area = 3 x 2 = 6 square units

8 by 6 rectangle

Area = Length x Width

Area = 8 x 6 = 48 square units

9 by 5 rectangle

Area = Length x Width

Area = 9 x 5 = 45 square units

Units for measuring area

We measure area using squares. We use different sizes of squares depending on how big or small an area is.

Example Length of side on Squares Unit

Size of the nail on your thumb

Millimeter

mm2

Size of piece of paper

Centimeter

cm2

Size of a room

Meter

m2

Size of a town

Kilometer

km2

 

Don't forget the wee 2

We write square sizes using a small 2 next to the unit.

We write mm2, cm2, m2 , km2, cm2

We can say "63 millimeters squared" or " 63 square millimeters"

We could use small squares to measure large areas. The only problem with this is that we would end up having to use very big numbers. For example, a field might be measured at 5,000,000,000 square millimeters when 5,000 square meters would be a much easier size to say, write, and visualize.

You will probably hear more units for measuring area; square inches, square feet, square yards, square miles, acres, hectares are all units used for measuring area.

More Examples of Calculating Area of Rectangles

9 by 4 mmm rectangle

Area = Length x Width

Area = 9 mm x 4 mm = 36 mm2

6 by 7 cm rectangle

Area = Length x Width

Area = 7 cm x 6 cm = 42 cm2

2 by 8 m rectangle

Area = Length x Width

Area = 8 m x 2 m = 16 m2

5 by 7 km rectangle

Area = Length x Width

Area = 7 km x 5 km = 35 km2

Area of a Square

The length and width of a square are the same so we just need to multiply the length by the length.

6 by 6 cm square

Area = Length x Length

Area = 6 cm x 6 cm = 36 cm2

Area of a Circle

The area of a circle = πr2
where r is the radius of the circle and π is the ratio of a circle's circumference to its diameter.

π (pronounced "pie" and often written "Pi") is an infinite decimal with a common approximation of 3.14159. You can find out more about Pi here

Example of calculating the area of a circle

circle with radius 4 cm marked

Area = πr2

Area = 3.14159 x (4 cm)2

Area = 3.14159 x 16 cm2

Area = 50.27 cm2

Answer rounded to 2 decimal places

Explanation of the Area of a Circle Formula

Take a circle and divide it into equally sized sectors and rearrange these as shown below. Notice how, as the sectors become smaller, the shape becomes more like a rectangle. Note: There is no limit to how small these sectors could be and to how closely they could resemble a rectangle when arranged.

Assuming we know that the circumference of a circle is equal to 2πr we can add dimensions to the "rectangle" as shown below. Using the area of a rectangle area formula, area = width x height we can see how our circle, re-configured as a rectangle, can be shown to have an area that approximates to πr x r or πr2

Circle Sectors Rearranged

circle shown with eight sectors arranged like a rectangle

Circle Sectors Rearranged - Starting to Look Like a Rectangle

circle shown with eight sectors arranged like a rectangle with height = radius and length = radius x pi

Area of Compound Shapes

There are many cases where the calculation of a total area requires more than one area to be calculated followed by either an addition, subtraction, or some other combination of operations to find the required area.

Note: In the examples below the units of measurement are not shown and answers and the value of π (Pi) have been rounded to the nearest hundredth.

Example: Simple Compound Shapes

The area calculation example below is relatively simple. The shape can be seen as a triangle combined with a rectangle.

compound shap comprising a triangle on top of a rectangle

Area of triangle part:
½ x base x height
½ x 9 x 4 = 18

Area of rectangle part:
width x height
9 x 6 = 54

Total area = 18 + 54 = 72

 

The example above illustrates a common requirement when working with compound shapes - finding dimensions that are not shown. When tutoring your children, give help, when needed, to find these "missing" dimensions. There is another example below.

Finding the dimensions

a shape made up of two rectangles with some dimensions shown

What are the dimensions of the small rectangular part?

Width? 12 - 7 - 2 = 3

Height? 8 - 6 = 2

 

Example: Subtracting one area from another

In the example below, the shape can be seen as a rectangle with a triangle cut out.

compoundd shape comprising a rectangle with a triangle cut out

Area of rectangle part:
width x height
5 x 6 = 30

Area of triangle part:
½ x base x height
½ x 3 x 3 = 4.50

Total area = 30 - 4.50 = 25.50

 

Example: Partial areas

The example below is similar to one above although, since we have a semi-circle we need to calculate a fraction (one-half) of the circle's area. Note in this example the diameter, and not the radius is shown.

compoundd shape comprising a triangle with a semi-circle cut out

Area of triangle part:
½ x base x height
½ x 6 x 6 = 18

Area of semi-circle part:
½ x (πr2)
½ x (3.14 x 1.52) = 3.53

Total area = 18 - 3.53 = 14.47

 

Example: Decisions! Combine? Subtract

It is common to have more than one way to calculate the final area. In the examples below the shape can be seen as two rectangles combined or as one large rectangle with a smaller rectangle "cut out" from the top right corner.

a compound shape comprising two rectangles

There is no right or wrong method.

a compound shape with area calculation method 1 of 2 shown

We can calculate the areas of rectangles a and b and then add to get the area.

area = (3 x 6) + (6 x 4)

area = 18 + 24 = 42

a compound shape with area calculation method 2 of 2 shown

Or we can calculate an "imaginary" rectangle, c, and subtract rectangle d from it to get the area.

area = (9 x 6) - (6 x 2)

area = 54 - 12 = 42

 

Calculating Area Worksheets

Print out the worksheets listed below and use them for practice when tutoring your children.

You will find more printable geometry worksheets here.

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