# Lesson 12

Volume of Right Prisms

### Lesson Narrative

In grades 5 and 6, students calculated the volume of rectangular prisms. In this lesson, students learn that they can calculate the volume of any right prism by multiplying the area of the base times the height of the prism. Students make sense of this formula by picturing the prism decomposed into identical layers 1 unit tall. These layers are composed of a number of cubic units equal to the number of square units in the area of the base. The height of the prism tells how many of these layers there are. Therefore, multiplying the number of cubic units in one layer times the number of layers gives the total number of cubic units in the prism, regardless of the shape of the base.

Given some three-dimensional figures that are prisms and some that are not, students decide whether they can apply the formula $$V = Bh$$ to calculate the volume. If so, they identify the base and measure the height, before calculating the volume. Students also apply the formula $$V = Bh$$ to find the height of a prism given its volume and the area of its base.

### Learning Goals

Teacher Facing

• Determine the volume of a right prism by counting how many unit cubes it takes to build one layer and then multiplying by the number of layers.
• Generalize (orally) the relationship between the volume of a prism, the area of its base, and its height.
• Identify whether a given figure is a prism, and if so, identify its base and height.

### Student Facing

Let’s look at volumes of prisms.

### Required Preparation

You will need the Finding Volume with Cubes blackline master for this lesson. You will only use one of the two pages. If your snap cubes measure $$\frac34$$ inch, print the first page of the blackline master, with the slightly smaller shapes. If your snap cubes measure 2 cm, print the second page of the blackline master, with the slightly larger shapes. Make sure to print the blackline master at 100% scale so the dimensions are accurate. Prepare 1 copy for every 6 students, and cut the pages in half so that each group of 3 students has one half-page.

Print, cut, and assemble the nets from the Can You Find the Volume? blackline master.  Card stock paper is recommended. Make sure to print the blackline master at 100% scale so the dimensions are accurate. Prepare 1 polyhedron for every 2 students (1 copy of the entire file for every 12--18 students).

Make sure students have access to snap cubes and rulers marked in centimeters.

### Student Facing

• I can explain why the volume of a prism can be found by multiplying the area of the base and the height of the prism.

### Glossary Entries

• base (of a prism or pyramid)

The word base can also refer to a face of a polyhedron.

A prism has two identical bases that are parallel. A pyramid has one base.

A prism or pyramid is named for the shape of its base.

• cross section

A cross section is the new face you see when you slice through a three-dimensional figure.

For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

• prism

A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

Here are some drawings of prisms.

• pyramid

A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

Here are some drawings of pyramids.

• volume

Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.