### Volume & capacity

Volume & Capacity |
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Assessment Questions |

The car can fit inside the container since there is **Space**

The amount of space occupied by an object is called its **volume**.

To measure volume we can use the following units:

- Cubic centimetre (cm³)
- Cubic metre (m³)

The amount that a container can hold is called **capacity**.

[resource: 2282, align: left]

The glass in the above image can hold a certain amount of liquid. This is called capacity.

Capacity can be measured using the units:

- Millilitres (ml)
- Decilitre (dl)
- Litre (l)

Relationships involving units of volume and capacity are given below.

** Volume **

1 000 000 cm³ = 1m³

**Capacity **

100 ml = 1 dl

10 dl = 1 l

1 000 ml = 1 l

**Volume and capacity**

1 cm³ = 1 ml

1 000 cm³ = 1

l 100 cm³= 1 dl

1 m³ = 1 000 l

Conversion and basic operations involving units of volume and capacity can be worked out using the relations given above.

To obtain the volume of cubes and cuboids, we can use the number of layers in a stack or apply formulae.

We can also use the relevant formulae to work out the volume of cylinders and solids with uniform cross-sections such as a triangular prism.

## Volume of a cube

A cube has 6 surfaces each of which is a square or all its edges (length, width and height) are

of the same length.

The volume is obtained by multiplying the 3 dimensions:

This is a Cuboid.

It has 6 faces which are not equal.

The opposite faces are equal to each other.

Volume of a cuboid = base area x height

Hence, ` l x w x h`

V = l × w × h

## A cylinder

#### Look at this image

A Cylinder has a **Top**, **Base** and a **Curved Surface**.

### Volume of a Cylinder

**Area of the base is the area of a circle** = πr^2

**Volume**= Area of the base x Height

#### Hence = πr^2× h

Play this video to see how to find the **Volume of a Cube**

Play this video to see how to find the **Volume of a Cylinder**

#### Example on Volume of a Cylinder

#### Volume of a Cuboid/Cube

#### Volume of a triangular prism

#### Volume of combined shapes

When we have cylinder , cube and cuboid together to form an object we call it a combined shape.

#### Example 1

Work out the volume of the below solid (Take π = 22 / 7)

#### Answer

The length, width and height of the cuboid are 15 m, 7 m and 8 m, respectively.

Also, the diameter of the half cylinder is 7 m and its height is 15 m.

So, the required volume = volume of the cuboid + 1/2 volume of the cylinder

#### Example 2

Work out the volume of the below solid (Take π = 22 / 7)

#### Answer

This solid is made up of a cuboid with a cylindrical hole.

To get its volume, Get volume of the cuboid (l x w x h) and Get the volume of the cylinder (pie x r x r x h)

Subtract the volume of the cylinder from the volume of the cuboid

Volume of the Cuboid = 16.2cm x 16.6cm x 24cm

** = 6298.56cm cubed**

Volume of the Cylinder = `22/7x 3.5cm x3.5cm x 24cm `

` `

** ** = 925cm cubed

Volume of the solid is therefore:

= (6298.56 - 924)cm cubed

= 5374.56 cm cubed.

#### Example 1

The bottom of a rectangular trough measures 150 cm by 350 cm.

If the height of the trough is 200 cm find its:

(a) volume in m³

(b) capacity in litres.

**Example 2**

A cylinder has a height of 200 cm. Its base radius is 50 cm.

Calculate its volume in m³. (Take ? = 3.14).

**Example 3**

A triangular prism has a volume of 360 cm³.

The prism has a length of 15 cm and the height of its triangular face is 8 cm.

Calculate the base of the triangular face.

## Attempt the examples above before moving onwards

#### Solution

**Example 1**

(a) **Volume** = length x width x height

= 150 cm x 350 cm x 200 cm

= 10 500 000 cm³

1 000 000 cm³ = 1 m³

Volume in m³

= 10 500 000 / 1 000 000

= 10.5 m³

(b) **Capacity**: 1 000 cm³ = 1 l

10 500 000 cm³ = l0 500 000/1 000 l

= 10 500 l

**Example 2**

**Solution **

Volume = pie r² h

= (3.14 x 50 x 50 x 200) cm³

= 1 570 000 cm³

1 000 000 cm³ = 1 m³

Volume = 1 570 000 / 1 000 000 (m³)

= 1.57 m³

#### Example 3

**Solution **

Let b represent the base of triangular face.

Volume = Area of cross-section x length

= Area of the triangular face x length

360 =1/2 x b x 8 x 15

b x 8 x 15 = 2 x 360

b = 2 x 360 / 8 x 15

= 6cm

## Capacity

The amount of liquid or gas that a container will hold when full. Or the volume of liquid that can fit inside the container

It measured in milliliters (ml), litre(l), deciliters(dl).

The standard unit used for capacity is litre (l).

Watch this video

Look at this too!

A cube with a volume of 1 cm3 will hold 1 ml of liquid

I ml = 1 cm3

1L = 1000 ml = 1000 cm3

.

The tank on a fuel tanker is in the shape of a cylinder 10 meters long, with a diameter of 3 metres.

a) Find to 2 decimal places the volume of the tank in cubic meters.

b) How many litres of fuel can this tank hold?

#### Answer

Volume =πr^2h

Take pie as 3.14

3.14×1.5×1.5×10

= 70.65m3

1M3= 1000L

70.65M3= ?

` "70.65M3" /"1M3" x1000L`

` `

=70,650L

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