E2.7 Show that the volume of a prism or cylinder can be determined by multiplying the area of its base by its height, and apply this relationship to find the area of the base, volume, and height of prisms and cylinders when given two of the three measurements.

Activity 1: Volume of a Right Rectangle-Based Prism


Goal

In this activity students realize that the volume of a prism is determined by the area of its base times its height.

Materials

  • various rectangle-based prisms (shoe box, cereal box, etc.).
  • many interlocking cubes of 1 cm by 1 cm.
Instructions

Form teams of two.

Give each team a prism and a fair amount of interlocking cubes.

Ask students to reproduce the prism with their cubes.

Ask the following questions:

  • What unit of measurement did you use to reproduce the prism?
  • How many cubic centimetres did it take to reproduce the length of the prism?
  • How many cubic centimetres did it take to reproduce the width?
  • How many cubic centimetres did it take to reproduce the height?
  • What are the dimensions of your prism?
  • What is the volume of your prism in cubic centimetres?
  • What is the relationship between the dimensions of your prism and its volume?
  • How can we determine the volume of any prism?

Sample answer for a 5 cm by 4 cm by 3 cm prism: "On the length, we can place five 1 cm3 units. On the width, we can place four 1 cm3 units, so we have a base of 5 × 4 units. On the height, we can place three 1 cm3 units. Therefore, the volume of the box is 5 × 4 × 3 units (cm3), or 60 cm3."

Note: Support students to realize that determining the volume of a prism involves finding the area of its base (5 × 4 units) and multiplying it by its height (3 units). Thus, the volume of a prism can be expressed as follows:

\(V = area \ of \ the \ base \times height \)

Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).

Activity 2: Volume of a Right Triangle-Based Prism


Goal

In this activity, students will determine the volume of a right triangle-based prism using the formula for calculating the volume of a right prism.

Materials

  • one ruler per team
  • a square or a goniometer per team
  • triangle-based prisms
  • blank paper, pencil and eraser.

Note: Coding software could be used to calculate the prism's volume from the formula.

Instructions

Form teams of two.

Give the teams a triangle-based prism.

Ask students to identify the bases of the prism.

Ask them to draw the outline of the prism's base on a piece of paper.

Using a square, they should then mark the height of the triangle and measure it in centimetres (2 cm).

For example:

Lead a class discussion with the following questions:

A prism with a triangular base.The measurement of the rectangular side is 5 centimeters by 8 centimeters.The height of the triangular base is 2 centimeters and the measurement of one of the sides is 5 centimeters.
  • What is the length of the triangle's base?
    • Expected answer: The length of the triangle's base is 5 cm.
  • What is the area of the triangle?
    • Expected answer: Since the area of a triangle is the base multiplied by the height and divided by 2, then the area of the triangle is \(\frac{5 \times 2}{2}\), which is 5. Therefore, the area of the triangle is 5 cm2, since the unit of area is cm2.
  • What is the height of the prism?
    • Expected answer: The height of the prism is 8 cm.
  • How is the volume of a prism determined?
    • Expected answer: The area of the base multiplied by its height.
  • What is the volume of your prism?
    • Expected answer: Since the height of the prism is 8 cm, then the volume is
      \(V = 5 cm^{2} \times 8 cm\)
      \(V = 40 cm^{3}\) Therefore, the volume of the prism is 40 cm3.
Enrichment Activity
  1. Find or construct a box shaped like a right triangle-based prism (for example, the box of a well-known Swiss chocolate brand).
  2. Determine the volume of the box.

Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).

Activity 3: Volume of Large Objects


Goal

In this activity, students will determine the volume of large objects using the most appropriate unit of volume measurement.

Materials

  • one metre per team
  • one cubic decimetre per team
  • a list of objects that are not all prisms (for example, a tree house, a hot water tank, a garbage can, a dictionary, the teacher's desk)
Instructions

Form teams of two.

Ask students to measure the lengths needed to determine the volume of the objects using the formula they discovered in the previous activities. Students can also construct an object that occupies approximately the same space.

Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).

Activity 4: Calculate the Height of a Prism from Its Area and Volume


Here are two three-dimensional objects, each with a volume of 126 dm3. Determine the height of each prism.

Rectangular prism whose side width is 7 decimetres and side length is 6 decimetres. The height is “h”.

A prism with a triangular base, the height is 'H'. The length of the base is 7 decimetres.

Source: translated from En avant, les maths!, 7e année, ML, Sens de l’espace, p. 21.

Activity 5: Calculating the Volume of a Cylinder


Find the volume of this cylinder.

A cylinder with a radius of 3 point 5 centimeters. The height of the cylinder is 8 centimeters.

A structure is formed by a right trapezoid-based prism topped by a cube. Determine the volume of the structure.

A right trapezoidal prism surmounted by a cube. All sides of the cube are equal and measure 5 meters. The height of the trapeze is 4 meters, its length is 11 meters, at the bottom and 5 meters at the top, and its width is 5 meters.

Source: translated from En avant, les maths!, 7e année, ML, Sens de l’espace, p. 22-23.

Activity 6: Area and Volume of Three-Dimensional Objects


Determine the area of the base, then the volume of each of the following prisms.

Prism with hexagonal base. The sides of the hexagon all measure 12 point 6 centimeters. The height of the prism is 8 centimeters. And the distance from the edge of the prism and the central point is ten point 8 centimeters.

A « house-shaped » pentagonal prism. The 5 sides of the pentagon are equal and measure 6 centimeters. The height is 8 centimeters.

A prism with a rectangular base. The base rectangle measures 8 centimeters by 5 centimeters. The height of the prism is ten centimeters.

A prism with a triangular base. The base triangle is a triangle with a right angle, the side opposite the right angle is ten centimeters. Another side of the triangle is 6 centimeters, the height is 5 centimeters.

Prism with hexagonal base. The sides of the hexagon all measure ten centimeters. The height of the prism is 5 centimeters. And the distance from the edge of the prism and the central point is 8 coma 66 centimeters.

Determine the area of the base and then the volume of each of the following cylinders.

Cylinder whose height is ten centimeters. The radius of the circular base is 5 centimeters.

Cylinder whose height is ten centimeters. The diameter of the circular base is ten centimeters.

Cylinder whose height is 8 centimeters. The circumference of the circular base is 20 centimeters.

Cylinder whose height is ten centimeters. The radius of the circular base is 4 point 2 centimeters.

The top of each end of the loaf below is shaped like a half circle. Determine the volume of the loaf. Show your work.

A prism that has the shape of a bread rounded on top, with a rectangular base of 11 centimeters by 20 centimeters. A dotted line determines the beginning of the semicircle. From the beginning of the semicircle to the rectangular base there are 6 centimeters.

A pipe measuring 300 cm long has an outside diameter of 8 cm and an inside diameter of 5 cm. How much material is needed to wrap it? Show your work.

A pipe whose diameter of the inner empty space has a diameter of 5 centimeters., the total diameter of the pipe is 8 centimeters. The height of the pipe is 2 decimeters.

Determine the interior volume of the smallest square-based prism box into which the cylinder below can be inserted.

A cylinder fits exactly into a square-based prism. The height of the cylinder is 50 centimeters. The radius of the circular base is 20 centimeters.

Source: translated from Réduction des écarts de rendement, 9e année, p. 10-11, 15-16.