092. Volume of Rectangular Prisms

Learning Intentions

• To understand what the volume of a three-dimensional object is
• To know that common metric units for volume include cubic millimetres, cubic centimetres, cubic metres and cubic kilometres
• To know what a rectangular prism (or cuboid) is
• find the volume of a cube and other rectangular prisms

Pre-requisite Summary

• Know that three-dimensional objects have length, width and height
• Know that volume measures the amount of space inside a three-dimensional object
• Know that volume is measured in cubic units such as ${\text{mm}}^3$, ${\text{cm}}^3$, ${\text{m}}^3$ and ${\text{km}}^3$
• Know that a cubic unit means a $1\times 1\times 1$ cube
• Know that a rectangular prism has $6$ rectangular faces
• Know that a cube is a special rectangular prism with all edges equal
• Know how to multiply whole numbers and decimals
• Know that the volume of a rectangular prism is found using $V=l\times w\times h$

Worked Examples

Worked Example 1

Decide whether each statement describes volume.
a) The amount of space inside a box
b) The distance around the edge of a shape
c) The amount of space a solid object takes up

Worked Example 2

Name the most suitable metric unit for the volume of each object.
a) A lunchbox
b) A shipping container
c) A tiny grain of sand

Worked Example 3

A rectangular prism has length $8\text{ cm}$, width $3\text{ cm}$ and height $5\text{ cm}$. Find its volume.

Worked Example 4

A cube has side length $6\text{ cm}$. Find its volume.

Worked Example 5

A rectangular prism has dimensions $1.2\text{ m}$, $0.5\text{ m}$ and $2\text{ m}$. Find its volume.

Worked Example 6

A storage box is $40\text{ cm}$ long, $25\text{ cm}$ wide and $30\text{ cm}$ high. Find its volume in cubic centimetres.

Problems

Problem 1

Decide whether each statement describes volume.
a) The amount of space inside a solid
b) The area of a flat surface
c) The space taken up by a 3D object

Problem 2

Name the most suitable metric unit for the volume of each object.
a) A classroom
b) A small dice
c) A very large mountain of rock

Problem 3

A rectangular prism has length $9\text{ cm}$, width $4\text{ cm}$ and height $7\text{ cm}$. Find its volume.

Problem 4

A cube has side length $5\text{ cm}$. Find its volume.

Problem 5

A rectangular prism has dimensions $1.5\text{ m}$, $0.4\text{ m}$ and $3\text{ m}$. Find its volume.

Problem 6

A fish tank is $60\text{ cm}$ long, $20\text{ cm}$ wide and $35\text{ cm}$ high. Find its volume in cubic centimetres.

Exercises

Understanding and Fluency

1. Decide whether each statement is about volume.
   a) The amount of space inside a solid
   b) The distance around a shape
   c) The number of cubic units needed to fill a solid

2. Choose the most suitable metric unit for volume.
   a) A tiny bead
   b) A shoebox
   c) A large building

3. Complete the statements.
   a) Volume is measured in $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ units
   b) A rectangular prism is a $\mathrm{\_}\mathrm{\_}\mathrm{\_}$-dimensional object
   c) A cube is a special type of $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ prism

4. Find the volume of each rectangular prism.
   a) $4\text{ cm}\times 3\text{ cm}\times 2\text{ cm}$
   b) $7\text{ cm}\times 5\text{ cm}\times 4\text{ cm}$
   c) $10\text{ m}\times 2\text{ m}\times 3\text{ m}$

5. Find the volume of each cube.
   a) Side length $3\text{ cm}$
   b) Side length $8\text{ mm}$
   c) Side length $1.5\text{ m}$

6. Find the volume of each rectangular prism.
   a) Length $12\text{ cm}$, width $6\text{ cm}$, height $4\text{ cm}$
   b) Length $9\text{ m}$, width $3\text{ m}$, height $2\text{ m}$
   c) Length $2.5\text{ cm}$, width $4\text{ cm}$, height $3\text{ cm}$

7. Find the missing dimension.
   a) $V=48{\text{ cm}}^3$, $l=4\text{ cm}$, $w=3\text{ cm}$
   b) $V=90{\text{ m}}^3$, $l=5\text{ m}$, $w=3\text{ m}$
   c) $V=64{\text{ cm}}^3$, cube side length $=?$

8. Decide whether each solid is a rectangular prism.
   a) A cube
   b) A cylinder
   c) A box-shaped brick

Reasoning

9. Explain why volume is measured in cubic units rather than square units.

10. A student says the volume of a rectangular prism is found by adding its length, width and height. Explain why this is incorrect.

11. Explain why a cube can be found using the same volume rule as a rectangular prism.

12. A rectangular prism has dimensions $6\text{ cm}$, $4\text{ cm}$ and $2\text{ cm}$. One student says the volume is $48{\text{ cm}}^3$ and another says it is $24{\text{ cm}}^2$. Determine who is correct and explain why.

Problem-solving

13. A gift box is $20\text{ cm}$ long, $15\text{ cm}$ wide and $10\text{ cm}$ high. Find its volume.

14. A storage crate is shaped like a cube with side length $40\text{ cm}$. Find its volume.

15. A fish tank is $80\text{ cm}$ long, $30\text{ cm}$ wide and $40\text{ cm}$ high. Find its volume.

16. A shipping container is $6\text{ m}$ long, $2.5\text{ m}$ wide and $2.4\text{ m}$ high. Find its volume.

17. A brick has dimensions $20\text{ cm}\times 10\text{ cm}\times 8\text{ cm}$. How many cubic centimetres does it occupy?

18. A cube-shaped container has volume $125{\text{ cm}}^3$. Find the length of one edge.

Potential Misunderstandings

• A student may confuse volume with area and think volume measures a flat surface
• A student may confuse volume with perimeter and think volume is about the distance around an object
• A student may use square units instead of cubic units
• A student may not recognise that a cube is a special rectangular prism
• A student may multiply only two dimensions instead of three
• A student may add the dimensions instead of multiplying them
• A student may omit the units or write incorrect units in the final answer
• A student may think cubic units are only used for cubes, rather than for all volume measurements

Next: 093. Volume of Triangular Prisms