# Volume Of A Liquid

1. Calculate the volume of a liquid in a rectangular container
2. Calculate the volume of liquid needed to fill a container to a certain height

### Volume recap

1. $\text{Volume of a cube} = \text{length} \times \text{length} \times \text{length}$

1. $\text{Volume of a cuboid} = \text{length} \times \text{breadth} \times \text{height}$

## 1. Calculate the volume of a liquid in a rectangular container

Question 1:

The diagram below shows a rectangular tank filled with some liquid.

Find the volume of the liquid in the tank.

Solution:

Length of cuboid $= \text{12 cm}$

Breadth of cuboid $= \text{5 cm}$

Height of liquid in cuboid $= \text{4 cm}$

Volume of a liquid in cuboid

\begin{align*}&= \mathrm{length \times breadth \times height} \\[2ex] &= \mathrm{12 \;cm \times 5 \;cm \times 4 \;cm} \\[2ex] &= \text{240 cm}^3 \end{align*}

$\text{240 cm}^3$

Question 2:

In the diagram below, find the volume of the liquid in the rectangular tank.

Solution:

Length of rectangular tank $= \text{14 m}$

Breadth of rectangular tank $= \text{5 m}$

Height of liquid in rectangular tank

\begin{align*}​ &= \text{10 m – 6 m} \\[2ex] &= \text{4 m} \end{align*}

Volume of liquid in tank

\begin{align*}&= \mathrm{length \times breadth \times height} \\[2ex] &= \mathrm{14 \;m \times 5 \;m \times 4 \;m} \\[2ex] &= \text{280 m}^3 \end{align*}

$\text{280 m}^3$

Question 3:

Find the volume of the liquid in the rectangular tank. Express your answer in litres.

Solution:

Length of rectangular tank $= \text{20 cm}$

Breadth of rectangular tank $= \text{8 cm}$

Height of liquid in rectangular tank

\begin{align*}​ &= \text{17 cm – 5 cm} \\[2ex] &= \text{12 cm} \end{align*}

Volume of liquid in the tank

\begin{align*}​ &= \mathrm{length \times breadth \times height} \\[2ex] &= \mathrm{20 \;cm \times 8 \;cm \times 12 \;cm} \\[2ex] &= \mathrm{1920 \;cm}^3 \\[2ex] &= \mathrm{1.92 \;litres} \end{align*}

$\text{1.92 litres}$

Question 4:

Chloe bought some apple juice and poured all of it into $\displaystyle{2}$ rectangular containers. Container $\displaystyle{\text{P}}$ measuring $\displaystyle{\text{15 cm}}$ by $\displaystyle{\text{8 cm}}$ by $\displaystyle{\text{6 cm}}$ is $\displaystyle{\frac{3}{4}}$ filled while Container $\displaystyle{\text{Q}}$ measuring $\displaystyle{\text{25 cm}}$ by $\displaystyle{\text{18 cm}}$ by $\displaystyle{\text{10 cm}}$ is $\displaystyle{\frac{5}{6}}$ filled. What is the volume of apple juice that Chloe bought?

Solution:

Volume of apple juice in Container $\displaystyle{\text{P}}$

\begin{align*}​ &= \mathrm{\frac{3}{4}\times \text{length} \times \text{breadth} \times \text{height}} \\[2ex] &= \mathrm{\frac{3}{4}\times 15 \;cm \times 8 \;cm \times 6 \;cm} \\[2ex] &= \mathrm{540 \;cm}^3 \\ \end{align*}

Volume of apple juice in Container $\displaystyle{\text{Q}}$

\begin{align*}​ &= \mathrm{\frac{5}{6}\times \text{length} \times \text{breadth} \times \text{height}} \\[2ex] &= \mathrm{\frac{5}{6}\times 25 \;cm \times 18 \;cm \times 10 \;cm} \\[2ex] &= \mathrm{3750 \;cm}^3 \\ \end{align*}

Total volume of apple juice Chloe bought

$=$ Volume of apple juice in Container $\displaystyle{\text{P + }}$ Volume of apple juice in Container $\displaystyle{\text{Q}}$

\begin{align*}​ &= \mathrm{540 \;cm}^3 + \mathrm{3750 \;cm}^3 \\[2ex] &= \mathrm{4290 \;cm}^3 \end{align*}

$\mathrm{4290 \;cm}^3$

Question 5:

A painter mixed some paint to paint a wall. He poured all the paint into two rectangular containers. Container $\text{X}$ measuring $\displaystyle{\text{55 cm}}$ by $\displaystyle{\text{48 cm}}$ by $\displaystyle{\text{30 cm}}$ was $\displaystyle{\frac{5}{8}}$ filled while Container $\text{Y}$ measuring $\displaystyle{\text{47 cm}}$ by $\displaystyle{\text{39 cm}}$ by $\displaystyle{\text{30 cm}}$ was $\displaystyle{\frac{10}{13}}$ filled. What was the volume of the paint that the painter mixed?

Solution:

Volume of paint in Container $\text{X}$

\begin{align*}​ &= \frac{5}{8}\times \text{length} \times \text{breadth} \times \text{height} \\[2ex] &= \frac{5}{8}\times 55 \text{ cm} \times 48 \text{ cm} \times 30 \text{ cm} \\[2ex] &= 49\,500 \text{ cm}^3 \\ \end{align*}

Volume of paint in Container $\text{Y}$

\begin{align*}​ &= \frac{10}{13}\times \text{length} \times \text{breadth} \times \text{height} \\[2ex] &= \frac{10}{13}\times 47 \text{ cm} \times 39 \text{ cm} \times 30 \text{ cm} \\[2ex] &= 42\,300 \text{ cm}^3 \\ \end{align*}

Total volume of paint mixed

$=$ Volume of paint in Container $\text{X}$ $+$ Volume of paint in Container $\text{Y}$

\begin{align*}​ &= 49 \,500 \text{ cm}^3 + 42 \,300 \text{ cm}^3 \\[2ex] &= 91 \,800 \text{ cm}^3 \end{align*}

$91 \,800 \text{ cm}^3$

## 2. Calculate the volume of liquid needed to fill a container to a certain height

Question 1:

A rectangular tank shown below was half–filled with water. Another $10.05 \text{ litres}$ of water was then added to the tank. How much more water was needed to fill the tank to the brim? Give your answer in litres.

Solution:

$1000 \text{ cm}^3 = 1 \mathrm{ \,\ell}$

Volume of water in the tank

\begin{align*}​ ​&= \frac{1}{2} \times 35 \text{ cm} \times 18 \text{ cm} \times 36 \text{ cm} \\[2ex] &= 11 \;340 \text{ cm}^3 \\[2ex] &= 11.34 \;ℓ​ \end{align*}

Remaining volume left to fill the tank

\begin{align*}​ ​&​= \frac{1}{2} \times 35 \text{ cm} \times 18 \text{ cm} \times 36 \text{ cm}\\[2ex] &= 11\,340 \text{ cm}^3\\[2ex] &= 11.34 \;ℓ​ \end{align*}

Volume of water added in the tank $= 10.05 \;ℓ$

Volume of water needed to fill till the brim

\begin{align*}​ ​ &​= 11.34 \;ℓ − 10.05 \;ℓ\\[2ex] &= 1.29 \;ℓ \end{align*}

$1.29 \;ℓ$

Question 2:

Jenny filled $\displaystyle{\frac {2}{3}}$ of a square–based container with a cocktail. The guests drank $\text{2.5 litres}$ of cocktail. How much more cocktail must she add in order to fill the container to the brim? Give your answer in litres.

Solution:

$1000 \text{ cm}^3 = 1 \;ℓ$

Volume of cocktail

\begin{align*}​ ​ &= \frac{1}{2} \times 11 \text{ cm} \times 11 \text{ cm} \times 45 \text{ cm} \\[2ex] &= 3630 \text{ cm}^3 \\[2ex] &= 3.63 \;ℓ​ \end{align*}

Volume of cocktail drank $= 2.5 \;ℓ$

Volume of cocktail left

$\displaystyle{ = 3.63 \;ℓ - 2.5 \;ℓ \\[2ex] = 1.13 \;ℓ }$

Volume of container

\begin{align}​ &= 11 \text{ cm} \times 11 \text{ cm} \times 45 \text{ cm} \\[2ex] &= 5445 \text{ cm}^3 \\[2ex] &= 5.445 \;ℓ \end{align}

Volume of cocktail needed to fill till the brim

\begin{align*}​ &= 5.445 \;ℓ - 1.13 \;ℓ \\[2ex] &= 4.315 \;ℓ \end{align*}

$4.315 \;ℓ$

Question 3:

Karen filled $\displaystyle{\frac {1}{3}}$ of rectangular Container $\text{A}$ measuring $\text{24 cm}$ by $\text{18 cm}$ by $\text{30 cm}$ with orange juice. Karen then added more orange juice such that half of Container $\text{A}$ was filled with orange juice. How much more orange juice did Karen add to Container $\text{A}$?

Solution:

Volume of orange juice in the tank

\begin{align*}​ ​&= \frac{1}{3} \times 24 \text{ cm} \times 18 \text{ cm} \times 30 \text{ cm} \\[2ex] &= 4320 \text{ cm}^3 \end{align*}

Volume of $\displaystyle{ \frac {1}{2}}$ of Container $\text{A}$

\begin{align*}​ ​ &= \frac{1}{2} \times 24 \text{ cm} \times 18 \text{ cm} \times 30 \text{ cm} \\[2ex] &= 6480 \text{ cm}^3 \end{align*}

Volume of orange juice Karen added

\begin{align}​ &= 6480 \text{ cm}^3 - 4320 \text{ cm}^3 \\[2ex] &= 2160 \text{ cm}^3 \end{align}

$2160 \text{ cm}^3$

OR

$\displaystyle{\frac {1}{2}-\frac{1}{3} = \frac{1}{6}}$

Volume of orange juice Karen added

\begin{align}​ ​\textstyle &= \frac{1}{6} \times 24 \text{ cm} \times 18 \text{ cm} \times 30 \text{ cm} \\[2ex] &= 2160 \text{ cm}^3 \\ \end{align}

$2160 \text{ cm}^3$

## Conclusion

In this article, we have learnt how to calculate the volume of liquids in a container as per the Primary 5 Maths level syllabus.

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