093. Volume of Triangular Prisms

Learning Intentions

• To understand that the volume of a prism can be calculated Use the area of the cross-section and the perpendicular height
• To know what a triangular prism is
• Solve 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 and
• 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 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
• 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 and perpendicular height . 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 and perpendicular height . The length of the prism is . Find its volume.

Worked Example 4

A triangular prism has a triangular cross-section with base and perpendicular height . The length of the prism is . Find its volume.

Worked Example 5

The triangular cross-section of a prism has area . The length of the prism is . Find the volume.

Worked Example 6

A tent is shaped like a triangular prism. Its triangular end has base and height . The tent is long. Find its volume.

Problems

Problem 1

A prism has a cross-section with area and perpendicular height . 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 and perpendicular height . The length of the prism is . Find its volume.

Problem 4

A triangular prism has a triangular cross-section with base and perpendicular height . The length of the prism is . Find its volume.

Problem 5

The triangular cross-section of a prism has area . The length of the prism is . Find the volume.

Problem 6

A greenhouse frame is shaped like a triangular prism. Its triangular end has base and height . The frame is long. Find its volume.

Exercises

Understanding and Fluency

Exercise 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 perpendicular height

c) A triangular prism has circular ends

Exercise 2.

Complete the statements.

a) The volume of a prism is

b) A triangular prism has cross-sections at its ends

c) Volume is measured in units

Exercise 3.

Find the volume of each prism.

a) Cross-sectional area , length

b) Cross-sectional area , length

c) Cross-sectional area , length

Exercise 4.

Find the area of each triangular cross-section, then the volume of the prism.

a) Base , triangle height , prism length

b) Base , triangle height , prism length

c) Base , triangle height , prism length

Exercise 5.

Find the volume of each triangular prism.

a) Triangle area , prism length

b) Triangle area , prism length

c) Triangle area , prism length

Exercise 6.

A triangular prism has the following measurements. Find the volume.

a) Base , triangle height , prism length

b) Base , triangle height , prism length

c) Base , triangle height , prism length

Exercise 7.

Find the missing length of the prism.

a) Volume , cross-sectional area

b) Volume , cross-sectional area

c) Volume , cross-sectional area

Exercise 8.

Find the missing area of the triangular cross-section.

a) Volume , prism length

b) Volume , prism length

c) Volume , prism length

Reasoning

Exercise 9.

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

Exercise 10.

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

Exercise 11.

Explain why the volume of a triangular prism can be written as .

Exercise 12.

A triangular prism has triangle base , triangle height and prism length . One student multiplies , while another uses . Determine who is correct and explain why.

Problem-solving

Exercise 13.

A tent is shaped like a triangular prism. Each triangular end has base and height . The tent is long. Find its volume.

Exercise 14.

A wedge-shaped block is a triangular prism with triangular cross-section base and height . Its length is . Find the volume.

Exercise 15.

A glasshouse is shaped like a triangular prism. Its triangular end has base and height . The glasshouse is long. Find its volume.

Exercise 16.

A triangular prism has volume . Its triangular cross-section has area . Find the length of the prism.

Exercise 17.

A triangular prism has volume and length . Find the area of its triangular cross-section.

Exercise 18.

A ramp is shaped like a triangular prism. Its side view is a triangle with base and height . The ramp is 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 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