**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.

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 m^{3} or 1,000,000 cm^{3}

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 cm^{3} in 1 m^{3} – 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 mm ^{3}, cm^{3}, m^{3} , km^{3}, cm^{3}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 cm ^{3} | |

Volume = Length x Width x Height Volume = 20 m x 2 m x 2 m = 80 m ^{3} | |

Volume = Length x Width x Height Volume = 10 m x 4 m x 5 m = 200 m ^{3} |

**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 = πr^{2} . There is more here on the area of a circle.

Volume = Area of base x Height Volume = πr ^{2} x hVolume = πr ^{2} 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 = πr^{2} hVolume = 3.14 x 3 x 3 x 8 Volume = 226.08 cm ^{3} |

**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 πr^{2}h |

**Example of Calculating the Volume of a Cone**

Dimensions are in cm. | Volume = 1/3 πr^{2} hVolume = 1/3 x 3.14 x 2 x 2 x 7 Volume = 29.31 cm ^{3} |

**Volume of a Sphere**

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

Volume = 4/3 πr^{3} |

**Example of Calculating the Volume of a Sphere**

Dimensions are in cm. | Volume = 4/3 πr^{3}Volume = 4/3 x 3.14 x 4 x 4 x 4 Volume = 267.95 cm ^{3} |

**Printable Volume Worksheets**

Use the worksheet below to practice calculating volumes.

- Finding Volume in Cubic Units – rectangular prisms and composites
- Volumes of Rectangular Prisms
- Volumes of Rectangular Prisms
- Volumes of “Real-world” objects e.g. of cereal boxes
- Calculating Volumes e.g. of triangular prisms and cylinders

You will get other geometry worksheets on perimeter, area, and more here.