The purpose of this activity is for students to connect their previous knowledge of the volume of rectangular prisms to their understanding of the volume of cylinders (MP7). From previous work, students should know that the volume of rectangular prisms is found by multiplying the area of the base by the height. Here we expand upon that to compute the volume of a cylinder.

Students start by calculating the volume of a rectangular prism. The goal is for students to then conjecture that the volume of a cylinder is the area of its base multiplied by its height. If some students don’t know that they should multiply the area of the base by its height, then they are prompted to connect prisms and cylinders to make a reasonable guess.

Use this discussion to highlight the important features of cylinders and their definitions: The radius of the cylinder is the radius of the circle that forms its base. The height of a cylinder is the length between its circular top and bottom. A cylinder of height 1 can be thought of as a “layer” in a cylinder with height . To highlight the connection between finding the area of a rectangular prism and finding the area of a cylinder, ask:

• “How are prisms and cylinders different?” (A prism has a base that is a polygon, and a cylinder has a base that is a circle.)
• “How are prisms and cylinders the same?” (The volume of cylinders and prisms is found by multiplying the area of the base by the height: )
• “How do you find the area of the base, , of a cylinder?” ().

In this activity, students use colored pencils (or pens or highlighters) to label the radius and height on different pictures of cylinders. Then they sketch their own cylinders and label the radius and heights of those. The purpose of this activity is for students to practice identifying the radius and height of various cylinders, some of which are in context.

This activity can also be abbreviated if students demonstrate prior understanding of how to draw or label cylinders and only need a brief refresh.

Select students to share where they marked the radius and height and the cylinders sketched for images E and F. Discuss examples of other cylinders students see in real life.

The purpose of this activity is for students to practice calculating volumes for cylinders. By finding the area of the base before finding the volume, students are encouraged to compute the volume by multiplying the area of its base by its height. This way of thinking about volume might be more intuitive for students than the formula .

The second problem of this activity focuses on exploring a cylinder in a context where a picture is not given and students are asked to make an label a sketch (MP4). When working with problems in a given context, it is sometimes convenient or practical to use an approximation of . An example of this is given in the question regarding the volume of a grain silo and interpretations of the answer.

The goal of this discussion is to ensure students understand how to use the area of the cylinder’s base to calculate its volume. Consider asking the following questions:

• “How does knowing the area of a circular base help determine the volume of a cylinder?” (The volume is this area multiplied by the height of the cylinder.)
• “If the cylinder were on its side, how do you know which measurements to use for the volume?” (It doesn’t matter which direction the cylinder is turned. We can always start by finding the radius or diameter of the circle to determine , and the height is always the distance between the bases.)
• “Do you prefer expressing your answer as a multiple of or as an approximation?” (I prefer answering with an approximation since I can judge the size of the number better when it’s written that way. Or, I prefer answering as a multiple of because I like having the exact value.)