127. Volume of Prisms and Cylinders

Learning Intentions

• To understand what a cross-section is of a prism and cylinder
• find the volume of a prism
• find the volume of a cylinder

Pre-requisite Summary

• Know that volume is the amount of space occupied by a three-dimensional object
• Know that volume is measured in cubic units such as ${\text{cm}}^3$, ${\text{m}}^3$ and ${\text{mm}}^3$
• Understand that area measures the surface inside a two-dimensional shape
• Be able to find the area of rectangles, triangles and circles
• Know that a prism has a constant cross-section along its length
• Know that a cylinder has circular cross-sections
• Be able to multiply decimals and whole numbers accurately
• Be able to substitute values into a formula

Worked Examples

Worked Example 1

State what the cross-section is in each solid:
a) a triangular prism
b) a rectangular prism
c) a cylinder

Worked Example 2

Find the volume of each prism using
$\text{Volume}=\text{area of cross-section}\times \text{length}$:
a) a rectangular prism with cross-section area $12{\text{ cm}}^2$ and length $8\text{ cm}$
b) a triangular prism with cross-section area $15{\text{ cm}}^2$ and length $6\text{ cm}$

Worked Example 3

Find the volume of each prism:
a) a rectangular prism with length $10\text{ cm}$, width $4\text{ cm}$ and height $3\text{ cm}$
b) a triangular prism with triangular cross-section of base $6\text{ cm}$ and height $5\text{ cm}$, and prism length $9\text{ cm}$

Worked Example 4

Find the volume of each cylinder using
$\text{Volume}=\pi r^{2}h$:
a) radius $=3\text{ cm}$, height $=10\text{ cm}$
b) diameter $=12\text{ cm}$, height $=7\text{ cm}$

Worked Example 5

Use $\pi \approx 3.14$ to find the volume of each cylinder:
a) radius $=5\text{ m}$, height $=8\text{ m}$
b) diameter $=14\text{ cm}$, height $=9\text{ cm}$

Worked Example 6

A solid has a constant cross-section. Find its volume:
a) cross-section area $=24{\text{ cm}}^2$, length $=11\text{ cm}$
b) a cylinder with base area $=50.24{\text{ cm}}^2$ and height $=6\text{ cm}$

Problems

Problem 1

State what the cross-section is in each solid:
a) a pentagonal prism
b) a cube
c) a cylinder

Problem 2

Find the volume of each prism using
$\text{Volume}=\text{area of cross-section}\times \text{length}$:
a) a prism with cross-section area $18{\text{ cm}}^2$ and length $7\text{ cm}$
b) a prism with cross-section area $20{\text{ cm}}^2$ and length $9\text{ cm}$

Problem 3

Find the volume of each prism:
a) a rectangular prism with length $12\text{ cm}$, width $5\text{ cm}$ and height $4\text{ cm}$
b) a triangular prism with triangular cross-section of base $8\text{ cm}$ and height $4\text{ cm}$, and prism length $10\text{ cm}$

Problem 4

Find the volume of each cylinder using
$\text{Volume}=\pi r^{2}h$:
a) radius $=4\text{ cm}$, height $=9\text{ cm}$
b) diameter $=16\text{ cm}$, height $=5\text{ cm}$

Problem 5

Use $\pi \approx 3.14$ to find the volume of each cylinder:
a) radius $=6\text{ m}$, height $=7\text{ m}$
b) diameter $=10\text{ cm}$, height $=12\text{ cm}$

Problem 6

A solid has a constant cross-section. Find its volume:
a) cross-section area $=32{\text{ cm}}^2$, length $=8\text{ cm}$
b) a cylinder with base area $=78.5{\text{ cm}}^2$ and height $=4\text{ cm}$

Exercises

Understanding and Fluency

1. Complete each statement:
   a) A cross-section is the shape made when a solid is cut ______ to its length
   b) A prism has the same cross-section all the way along its ______
   c) The cross-section of a cylinder is a ______
   d) Volume is measured in ______ units

2. State the cross-section of each solid:
   a) triangular prism
   b) rectangular prism
   c) hexagonal prism
   d) cylinder

3. Find the volume of each prism using cross-section area $\times$ length:
   a) cross-section area $=10{\text{ cm}}^2$, length $=9\text{ cm}$
   b) cross-section area $=14{\text{ m}}^2$, length $=5\text{ m}$
   c) cross-section area $=7.5{\text{ cm}}^2$, length $=8\text{ cm}$

4. Find the volume of each rectangular prism:
   a) $8\text{ cm}\times 3\text{ cm}\times 2\text{ cm}$
   b) $15\text{ m}\times 4\text{ m}\times 6\text{ m}$
   c) $12\text{ mm}\times 5\text{ mm}\times 7\text{ mm}$

5. Find the volume of each triangular prism:
   a) triangle base $=6\text{ cm}$, triangle height $=4\text{ cm}$, prism length $=10\text{ cm}$
   b) triangle base $=9\text{ m}$, triangle height $=5\text{ m}$, prism length $=7\text{ m}$
   c) triangle base $=8\text{ cm}$, triangle height $=3\text{ cm}$, prism length $=12\text{ cm}$

6. Find the volume of each cylinder using $\pi \approx 3.14$:
   a) radius $=3\text{ cm}$, height $=8\text{ cm}$
   b) radius $=7\text{ m}$, height $=4\text{ m}$
   c) diameter $=10\text{ cm}$, height $=6\text{ cm}$

7. Find the volume of each cylinder using a calculator:
   a) radius $=5\text{ cm}$, height $=9\text{ cm}$
   b) diameter $=14\text{ cm}$, height $=12\text{ cm}$
   c) radius $=2.5\text{ m}$, height $=11\text{ m}$

8. Solve each:
   a) A prism has cross-section area $18{\text{ cm}}^2$ and length $13\text{ cm}$. Find its volume
   b) A cylinder has base area $28.26{\text{ cm}}^2$ and height $10\text{ cm}$. Find its volume
   c) A rectangular prism has volume $240{\text{ cm}}^3$, width $4\text{ cm}$ and height $5\text{ cm}$. Find its length

Reasoning

9. Explain why the volume of a prism can be found by multiplying the area of its cross-section by its length.

10. A student says that the cross-section of a cylinder is a rectangle. Explain the mistake.

11. Noah says that the volume of a cylinder is found using $2\pi rh$. Is he correct? Explain.

12. Explain why the diameter must be halved before using the cylinder volume formula if the formula uses $r$.

13. A student finds the volume of a triangular prism by using $\text{base}\times \text{height}\times \text{length}$ for the triangle. Describe the error.

Problem-solving

14. A juice box is a rectangular prism with length $6\text{ cm}$, width $4\text{ cm}$ and height $10\text{ cm}$. Find its volume.

15. A tent is shaped like a triangular prism. The triangular cross-section has base $3\text{ m}$ and height $2\text{ m}$, and the tent is $5\text{ m}$ long. Find its volume.

16. A can is shaped like a cylinder with radius $4\text{ cm}$ and height $12\text{ cm}$. Find its volume using $\pi \approx 3.14$.

17. A cylinder has diameter $20\text{ cm}$ and height $15\text{ cm}$. Find its volume using $\pi \approx 3.14$.

18. A prism has constant cross-section area $25{\text{ cm}}^2$ and length $14\text{ cm}$. Find its volume.

19. A solid has the shape of a cylinder with base area $153.86{\text{ cm}}^2$ and height $7\text{ cm}$. Find its volume.

Potential Misunderstandings

• Students may confuse cross-section with a face that is not the repeated shape
• Students may think any cut through a prism gives the standard cross-section
• Students may confuse area and volume
• Students may use square units instead of cubic units for volume
• Students may forget to find the area of the triangular cross-section before multiplying by prism length
• Students may use the diameter instead of the radius in the cylinder formula
• Students may forget to square the radius in $\pi r^{2}h$
• Students may use circumference formulas when finding cylinder volume
• Students may think the volume formula for a cylinder is unrelated to the prism formula, rather than using base area $\times$ height