126. Volume and Capacity of Prisms

Learning Intentions

• To understand that volume is the space occupied by a three-dimensional object
• To understand that capacity is the volume of fluid or gas that a container can hold
• convert between units for volume and capacity
• find the volume of rectangular prisms, including cubes

Pre-requisite Summary

• Know that three-dimensional objects have length, width and height
• Understand that area measures the amount of surface inside a two-dimensional shape
• Know that volume measures space in three dimensions
• Understand that volume is measured in cubic units such as ${\text{cm}}^3$ and ${\text{m}}^3$
• Understand that capacity is commonly measured in mL, L and kL
• Know basic metric conversions such as $1\text{ L}=1000\text{ mL}$
• Know that a cube is a special rectangular prism with all edges equal
• Be able to multiply whole numbers and decimals accurately

Worked Examples

Worked Example 1

State the meaning of each term and identify a suitable unit:
a) volume of a box
b) capacity of a water bottle
c) capacity of a swimming pool

Worked Example 2

Convert between units for volume and capacity:
a) $2500\text{ mL}$ to L
b) $3.5\text{ L}$ to mL
c) $2\text{ kL}$ to L

Worked Example 3

Convert between units for volume and capacity:
a) $4000{\text{ cm}}^3$ to L
b) $1.8\text{ L}$ to ${\text{cm}}^3$
c) $0.75{\text{ m}}^3$ to L

Worked Example 4

Find the volume of each rectangular prism:
a) length $=8\text{ cm}$, width $=3\text{ cm}$, height $=5\text{ cm}$
b) length $=12\text{ m}$, width $=4\text{ m}$, height $=2\text{ m}$

Worked Example 5

Find the volume of each cube:
a) side length $=6\text{ cm}$
b) side length $=1.5\text{ m}$

Worked Example 6

A fish tank is a rectangular prism with length $50\text{ cm}$, width $30\text{ cm}$ and height $40\text{ cm}$.
a) Find its volume in ${\text{cm}}^3$
b) Convert this volume to litres

Problems

Problem 1

State the meaning of each term and identify a suitable unit:
a) volume of a cupboard
b) capacity of a milk carton
c) capacity of a dam

Problem 2

Convert between units for volume and capacity:
a) $1800\text{ mL}$ to L
b) $4.2\text{ L}$ to mL
c) $5\text{ kL}$ to L

Problem 3

Convert between units for volume and capacity:
a) $2500{\text{ cm}}^3$ to L
b) $2.4\text{ L}$ to ${\text{cm}}^3$
c) $1.2{\text{ m}}^3$ to L

Problem 4

Find the volume of each rectangular prism:
a) length $=9\text{ cm}$, width $=4\text{ cm}$, height $=7\text{ cm}$
b) length $=10\text{ m}$, width $=3\text{ m}$, height $=6\text{ m}$

Problem 5

Find the volume of each cube:
a) side length $=4\text{ cm}$
b) side length $=2\text{ m}$

Problem 6

A storage container is a rectangular prism with length $60\text{ cm}$, width $25\text{ cm}$ and height $20\text{ cm}$.
a) Find its volume in ${\text{cm}}^3$
b) Convert this volume to litres

Exercises

Understanding and Fluency

1. Complete each statement:
   a) Volume is the amount of ______ occupied by a 3D object
   b) Capacity is the amount of fluid or gas a ______ can hold
   c) Volume is measured in ______ units
   d) Capacity is often measured in ______ and ______

2. State whether each measure is volume or capacity:
   a) the amount of space inside a cardboard box
   b) the amount of juice a bottle can hold
   c) the amount of space occupied by a brick
   d) the amount of water a tank can hold

3. Convert between units for capacity:
   a) $3500\text{ mL}$ to L
   b) $6\text{ L}$ to mL
   c) $2.5\text{ kL}$ to L
   d) $750\text{ mL}$ to L

4. Convert between units for volume and capacity:
   a) $5000{\text{ cm}}^3$ to L
   b) $3.2\text{ L}$ to ${\text{cm}}^3$
   c) $2{\text{ m}}^3$ to L
   d) $0.4{\text{ m}}^3$ to L

5. Find the volume of each rectangular prism:
   a) $7\text{ cm}\times 4\text{ cm}\times 3\text{ cm}$
   b) $12\text{ m}\times 5\text{ m}\times 2\text{ m}$
   c) $9\text{ mm}\times 6\text{ mm}\times 8\text{ mm}$

6. Find the volume of each cube:
   a) side length $5\text{ cm}$
   b) side length $9\text{ mm}$
   c) side length $1.2\text{ m}$

7. Find the volume, then convert where needed:
   a) a prism $10\text{ cm}\times 8\text{ cm}\times 5\text{ cm}$, answer in ${\text{cm}}^3$ and L
   b) a cube with side $10\text{ cm}$, answer in ${\text{cm}}^3$ and L
   c) a prism $2\text{ m}\times 1\text{ m}\times 0.5\text{ m}$, answer in ${\text{m}}^3$ and L

8. Solve each:
   a) A rectangular prism has length $15\text{ cm}$, width $6\text{ cm}$ and height $4\text{ cm}$. Find its volume
   b) A cube has volume $64{\text{ cm}}^3$. Find its side length
   c) A container has capacity $1.5\text{ L}$. Write this in mL and ${\text{cm}}^3$

Reasoning

9. Explain why volume is measured in cubic units instead of square units.

10. A student says that the volume of a rectangular prism is found by adding length, width and height. Explain the mistake.

11. Noah says that $1\text{ L}=100{\text{ cm}}^3$. Is he correct? Explain.

12. Explain why capacity and volume are closely related but not exactly the same idea.

13. A student says that a cube with side length $3\text{ cm}$ has volume $9{\text{ cm}}^3$. Describe the error.

Problem-solving

14. A lunchbox measures $20\text{ cm}$ by $12\text{ cm}$ by $8\text{ cm}$. Find its volume.

15. A carton has capacity $2\text{ L}$. Write this capacity in mL and ${\text{cm}}^3$.

16. A fish tank measures $80\text{ cm}$ by $35\text{ cm}$ by $50\text{ cm}$. Find its volume in ${\text{cm}}^3$ and litres.

17. A cube-shaped container has side length $25\text{ cm}$. Find its volume.

18. A swimming pool contains $40{\text{ m}}^3$ of water. Convert this to litres.

19. A storage box has volume $960{\text{ cm}}^3$. Its length is $12\text{ cm}$ and its width is $8\text{ cm}$. Find its height.

Potential Misunderstandings

• Students may confuse volume with area or perimeter
• Students may think capacity and volume mean exactly the same thing in every context
• Students may use square units instead of cubic units for volume
• Students may forget that $1\text{ L}=1000\text{ mL}$
• Students may forget that $1\text{ L}=1000{\text{ cm}}^3$
• Students may forget that $1{\text{ m}}^3=1000\text{ L}$
• Students may add dimensions instead of multiplying them when finding volume
• Students may think the volume of a cube is found by doubling or squaring its side length instead of cubing it
• Students may confuse the units when converting between volume and capacity
• Students may not check whether their final answer is reasonable for the size of the object