# How to Calculate the Volume

## What is volume?

Volume measures how much space an object occupies. Sometimes you might hear questions like "what is the capacity of a box?" or "how much can the box hold?" You can assume that these questions will need a volume to be calculated.
Note: To be totally smart, volume and capacity aren't always the same - think of a box with really thick sides!

## Calculating Volume

Volume is measured in cubes (or cubic units).

How many cubes are in this rectangular prism (cuboid)?

We can count the cubes although it is quicker to take the length, width, and height and use multiplication. The rectangular prism above has an volume of 48 cubic units.

The volume of a rectangular prism is = length x width x height

### Examples of calculating the area of a rectangle

We need to do two multiplications to work out the volume. We calculate the area of one face (or side) and multiply that by its height. The examples below show how there are three ways of doing this.

 Area = 6 x 4 = 24 Volume = Area x 2 Volume = 24 x 2 = 48 cubic units Area = 6 x 2 = 12 Volume = Area x 4 Volume = 12 x 4 = 48 cubic units Area = 4 x 2 = 8 Volume = Area x 6 Volume = 8 x 6 = 48 cubic units

Notice how we get the same answer no matter what side we use to find an area.

When your child starts working with area and perimeter he or she will usually work with 2 dimensions - squares, rectangles, triangles, etc. that are shown on paper as flat - there is no depth, or 3rd dimension. Working with volume does involve 3 dimensions. Ensure your child is aware of this and does not think of the cubes, and other 3D shapes shown on paper as just being another "shape on the page." Show them real boxes, and show how these can be drawn (or represented) on a two dimensional piece of paper. In other words, make sure the connection between what's on paper and what it represents in the real world is made.

Be sure your child is not confused by the use of volume as used when talking about loudness.

## Units for measuring volume

There are very big differences between units of measurement for volume. For example, there are 100 centimeters in 1 meter but there are 1,000,000 (yes, 1 million) cubic centimeters in a cubic meter.

Why the big difference? Because in volume we have not just length; we have length, width, and height. The sugar cube example below shows this.

How much sugar? 1 m3 or 1,000,000 cm3

 Think of filling a very big box (it would be 1 meter wide, 1 meter, long, and one meter high) with sugar cubes (with each side 1 centimeter). Step 1: one row along the bottom of the box - that would be 100 sugar cubes Step 2: cover the rest of the base of the box - that would give a total of 100 rows each with 100 sugar cubes. 100 x 100 = 10,000 sugar cubes at the bottom of the big box. Step 3: Repeat this 99 times until there are layers of 10,000 cubes stacked 100 deep. 10,000 x 100 = 1,000,000 sugar cubes

There are 1,000,000 cm3 in 1 m3 - be careful not to have too much sugar!

There are other units for measuring volume; cubic inches, cubic feet, cubic yards are all units used for measuring volume. Milliliters, liters, gallons are also used especially when measuring liquids.

 Don't forget the wee 3 We write cubic sizes using a small 3 next to the unit. We write mm3, cm3, m3 , km3, cm3 We can say "85 centimeters cubed" or "85 cubic centimeters"

## Examples of Calculating Volume of Rectangular Prisms

 Volume = Length x Width x Height Volume = 12 cm x 8 cm x 6 cm = 576 cm3 Volume = Length x Width x Height Volume = 20 m x 2 m x 2 m = 80 m3 Volume = Length x Width x Height Volume = 10 m x 4 m x 5 m = 200 m3

## Volume of a Cylinder

Calculating the volume of a cylinder involves multiplying the area of the base by the height of the cylinder. The base of a cylinder is circular and the formula for the area of a circle is: area of a circle = πr2 . There is more here on the area of a circle.

 Volume = Area of base x Height Volume = πr2 x h Volume = πr2 h

Note: in the examples below we will use 3.14 as an approximate value for π (Pi).

### Example of Calculating the Volume of a Cylinder

 Dimensions are in cm. Volume = πr2 h Volume = 3.14 x 3 x 3 x 8 Volume = 226.08 cm3

## Volume of a Cone

The volume of a cone is equal to one-third the volume of a cylinder with matching height and area of base. This gives the formula for the volume of a cone as shown below.

 Volume = 1/3 πr2h

### Example of Calculating the Volume of a Cone

 Dimensions are in cm. Volume = 1/3 πr2 h Volume = 1/3 x 3.14 x 2 x 2 x 7 Volume = 29.31 cm3

## Volume of a Sphere

The formula for the volume of a sphere is shown below.

 Volume = 4/3 πr3

### Example of Calculating the Volume of a Sphere

 Dimensions are in cm. Volume = 4/3 πr3 Volume = 4/3 x 3.14 x 4 x 4 x 4 Volume = 267.95 cm3

## Printable Volume Worksheets

Use the worksheet below to practice calculating volumes.

All Geometry 24 Worksheets Terms/ Definitions Formulas/ Equations 2D Shapes 3D Shapes Quadrilaterals Measuring Angles Using a Protractor Adding and Subtracting Angles Angle Properties Finding Angles Symmetry Area Volume Surface Area Perimeter Coordinate System Coordinate Graphing Pythagoras' Theorem Distance Between Two Points Congruent Triangles Similar Triangles Transformations Dilations
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