127. Volume of Prisms and Cylinders

Learning Intentions

• To understand what a cross-section is of a prism and cylinder
• Solve 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 , and
• 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 Use

:

a) a rectangular prism with cross-section area and length

b) a triangular prism with cross-section area and length

Worked Example 3

Find the volume of each prism:

a) a rectangular prism with length , width and height

b) a triangular prism with triangular cross-section of base and height , and prism length

Worked Example 4

Find the volume of each cylinder using

:

a) radius , height

b) diameter , height

Worked Example 5

Use to find the volume of each cylinder:

a) radius , height

b) diameter , height

Worked Example 6

A solid has a constant cross-section. Find its volume:

a) cross-section area , length

b) a cylinder with base area and height

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

:

a) a prism with cross-section area and length

b) a prism with cross-section area and length

Problem 3

Find the volume of each prism:

a) a rectangular prism with length , width and height

b) a triangular prism with triangular cross-section of base and height , and prism length

Problem 4

Find the volume of each cylinder using

:

a) radius , height

b) diameter , height

Problem 5

Use to find the volume of each cylinder:

a) radius , height

b) diameter , height

Problem 6

A solid has a constant cross-section. Find its volume:

a) cross-section area , length

b) a cylinder with base area and height

Exercises

Understanding and Fluency

Exercise 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

Exercise 2.

State the cross-section of each solid:

a) triangular prism

b) rectangular prism

c) hexagonal prism

d) cylinder

Exercise 3.

Find the volume of each prism using cross-section area length:

a) cross-section area , length

b) cross-section area , length

c) cross-section area , length

Exercise 4.

Find the volume of each rectangular prism:

a)

b)

c)

Exercise 5.

Find the volume of each triangular prism:

a) triangle base , triangle height , prism length

b) triangle base , triangle height , prism length

c) triangle base , triangle height , prism length

Exercise 6.

Find the volume of each cylinder using :

a) radius , height

b) radius , height

c) diameter , height

Exercise 7.

Find the volume of each cylinder using a calculator:

a) radius , height

b) diameter , height

c) radius , height

Exercise 8.

Solve each:

a) A prism has cross-section area and length . Find its volume

b) A cylinder has base area and height . Find its volume

c) A rectangular prism has volume , width and height . Find its length

Reasoning

Exercise 9.

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

Exercise 10.

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

Exercise 11.

Noah says that the volume of a cylinder is found using . Is he correct? Explain.

Exercise 12.

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

Exercise 13.

A student finds the volume of a triangular prism by using for the triangle. Describe the error.

Problem-solving

Exercise 14.

A juice box is a rectangular prism with length , width and height . Find its volume.

Exercise 15.

A tent is shaped like a triangular prism. The triangular cross-section has base and height , and the tent is long. Find its volume.

Exercise 16.

A can is shaped like a cylinder with radius and height . Find its volume using .

Exercise 17.

A cylinder has diameter and height . Find its volume using .

Exercise 18.

A prism has constant cross-section area and length . Find its volume.

Exercise 19.

A solid has the shape of a cylinder with base area and height . 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
• 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 height