093. Volume of Triangular Prisms

Learning Intentions

• To understand that the volume of a prism can be calculated using the area of the cross-section and the perpendicular height
• To know what a triangular prism is
• find the volume of a triangular prism

Pre-requisite Summary

• Know that volume measures the amount of space inside a three-dimensional object
• Know that volume is measured in cubic units such as ${\text{cm}}^3$ and ${\text{m}}^3$
• Know that a prism has a constant cross-section all the way through
• Know that the volume of a prism can be found using area of cross-section $\times$ perpendicular height (or length of the prism)
• Know that a triangular prism has triangular ends and rectangular side faces
• Know how to find the area of a triangle using $A={\displaystyle \frac{1}{2}}bh$
• Know that the perpendicular height of a prism is the distance between the matching cross-sections

Worked Examples

Worked Example 1

A prism has a cross-section with area $12{\text{ cm}}^2$ and perpendicular height $8\text{ cm}$. Find its volume.

Worked Example 2

Identify the shape as a triangular prism or not a triangular prism.
a) A solid with two parallel triangular faces and three rectangular faces
b) A solid with two circular faces and one curved surface
c) A solid with two parallel rectangular faces and four rectangular faces

Worked Example 3

A triangular prism has a triangular cross-section with base $6\text{ cm}$ and perpendicular height $4\text{ cm}$. The length of the prism is $10\text{ cm}$. Find its volume.

Worked Example 4

A triangular prism has a triangular cross-section with base $9\text{ m}$ and perpendicular height $5\text{ m}$. The length of the prism is $7\text{ m}$. Find its volume.

Worked Example 5

The triangular cross-section of a prism has area $18{\text{ cm}}^2$. The length of the prism is $12\text{ cm}$. Find the volume.

Worked Example 6

A tent is shaped like a triangular prism. Its triangular end has base $2.4\text{ m}$ and height $1.5\text{ m}$. The tent is $3\text{ m}$ long. Find its volume.

Problems

Problem 1

A prism has a cross-section with area $15{\text{ cm}}^2$ and perpendicular height $9\text{ cm}$. Find its volume.

Problem 2

Identify the shape as a triangular prism or not a triangular prism.
a) A solid with two parallel triangular faces and three rectangular faces
b) A solid with one triangular face and one square face
c) A solid with two parallel triangular faces joined by rectangles

Problem 3

A triangular prism has a triangular cross-section with base $8\text{ cm}$ and perpendicular height $3\text{ cm}$. The length of the prism is $11\text{ cm}$. Find its volume.

Problem 4

A triangular prism has a triangular cross-section with base $10\text{ m}$ and perpendicular height $6\text{ m}$. The length of the prism is $8\text{ m}$. Find its volume.

Problem 5

The triangular cross-section of a prism has area $24{\text{ cm}}^2$. The length of the prism is $14\text{ cm}$. Find the volume.

Problem 6

A greenhouse frame is shaped like a triangular prism. Its triangular end has base $3\text{ m}$ and height $2\text{ m}$. The frame is $5\text{ m}$ long. Find its volume.

Exercises

Understanding and Fluency

1. Decide whether each statement is true or false.
   a) A prism has the same cross-section all the way through
   b) The volume of a prism can be found using cross-sectional area $\times$ perpendicular height
   c) A triangular prism has circular ends

2. Complete the statements.
   a) The volume of a prism is $V=\mathrm{\_}\mathrm{\_}\mathrm{\_}$
   b) A triangular prism has $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ cross-sections at its ends
   c) Volume is measured in $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ units

3. Find the volume of each prism.
   a) Cross-sectional area $7{\text{ cm}}^2$, length $5\text{ cm}$
   b) Cross-sectional area $12{\text{ cm}}^2$, length $9\text{ cm}$
   c) Cross-sectional area $20{\text{ m}}^2$, length $4\text{ m}$

4. Find the area of each triangular cross-section, then the volume of the prism.
   a) Base $6\text{ cm}$, triangle height $4\text{ cm}$, prism length $10\text{ cm}$
   b) Base $10\text{ cm}$, triangle height $3\text{ cm}$, prism length $8\text{ cm}$
   c) Base $8\text{ m}$, triangle height $5\text{ m}$, prism length $6\text{ m}$

5. Find the volume of each triangular prism.
   a) Triangle area $15{\text{ cm}}^2$, prism length $7\text{ cm}$
   b) Triangle area $24{\text{ cm}}^2$, prism length $12\text{ cm}$
   c) Triangle area $9{\text{ m}}^2$, prism length $5\text{ m}$

6. A triangular prism has the following measurements. Find the volume.
   a) Base $12\text{ cm}$, triangle height $5\text{ cm}$, prism length $9\text{ cm}$
   b) Base $7\text{ cm}$, triangle height $6\text{ cm}$, prism length $4\text{ cm}$
   c) Base $9\text{ m}$, triangle height $4\text{ m}$, prism length $10\text{ m}$

7. Find the missing length of the prism.
   a) Volume $120{\text{ cm}}^3$, cross-sectional area $15{\text{ cm}}^2$
   b) Volume $84{\text{ m}}^3$, cross-sectional area $12{\text{ m}}^2$
   c) Volume $150{\text{ cm}}^3$, cross-sectional area $25{\text{ cm}}^2$

8. Find the missing area of the triangular cross-section.
   a) Volume $96{\text{ cm}}^3$, prism length $8\text{ cm}$
   b) Volume $135{\text{ m}}^3$, prism length $9\text{ m}$
   c) Volume $72{\text{ cm}}^3$, prism length $6\text{ cm}$

Reasoning

9. Explain why the volume of any prism can be found by multiplying the area of the cross-section by the perpendicular height.

10. A student says a triangular prism has a triangle on only one end. Explain why this is incorrect.

11. Explain why the volume of a triangular prism can be written as $V={\displaystyle \frac{1}{2}}bh\times l$.

12. A triangular prism has triangle base $8\text{ cm}$, triangle height $5\text{ cm}$ and prism length $12\text{ cm}$. One student multiplies $8\times 5\times 12$, while another uses ${\displaystyle \frac{1}{2}}\times 8\times 5\times 12$. Determine who is correct and explain why.

Problem-solving

13. A tent is shaped like a triangular prism. Each triangular end has base $2\text{ m}$ and height $1.5\text{ m}$. The tent is $4\text{ m}$ long. Find its volume.

14. A wedge-shaped block is a triangular prism with triangular cross-section base $10\text{ cm}$ and height $6\text{ cm}$. Its length is $15\text{ cm}$. Find the volume.

15. A glasshouse is shaped like a triangular prism. Its triangular end has base $5\text{ m}$ and height $3\text{ m}$. The glasshouse is $8\text{ m}$ long. Find its volume.

16. A triangular prism has volume $180{\text{ cm}}^3$. Its triangular cross-section has area $20{\text{ cm}}^2$. Find the length of the prism.

17. A triangular prism has volume $210{\text{ m}}^3$ and length $7\text{ m}$. Find the area of its triangular cross-section.

18. A ramp is shaped like a triangular prism. Its side view is a triangle with base $1.8\text{ m}$ and height $0.6\text{ m}$. The ramp is $2.5\text{ m}$ wide. Find the volume.

Potential Misunderstandings

• A student may confuse the height of the triangular cross-section with the length of the prism
• A student may think the prism formula uses the sloping side of the triangle instead of the perpendicular triangle height
• A student may forget to find the area of the triangular cross-section before multiplying by the prism length
• A student may use the formula for a rectangular prism instead of the formula for a triangular prism
• A student may think a triangular prism has only one triangular face
• A student may confuse square units and cubic units
• A student may omit the factor of ${\displaystyle \frac{1}{2}}$ when finding the area of the triangular cross-section
• A student may not recognise that the cross-section must stay the same all the way through the prism

Next: 094r. Capacity and Volume Conversions