In this activity, students revisit how to calculate the volume of a cone, sphere, and cylinder. They apply volume formulas, solve for the radius, and analyze the relationship between a solid’s dimensions and its volume. Students look for and make use of structure (MP7) while using formulas to calculate the volume and find unknown dimensions of different solids.

The purpose of this discussion is to help students notice patterns in volume calculations across different three-dimensional solids. Display the 3 solids and their volume formulas for all to see. Here are some questions for discussion:

• “How do the volumes of the cylinder and cone compare if they have the same radius and height?” (The volume of the cylinder is 3 times the volume of the cone.)
• “What was similar and different about finding the volume of each solid versus finding the radius of each solid?” (In both calculations, the volume formula is used. Solving for the radius requires taking a square root or a cube root.)

If students are unsure of how to find the volume of a hemisphere, consider asking:

• “What do you notice about the shape of the hemisphere?”
• “What is the same and different about a sphere and a hemisphere?”

The goal is to make sure students understand how to break composite figures into familiar three-dimensional solids to find the volume. Begin by inviting 2–4 previously selected students to share their sketches and how they got started determining the volumes. In particular, focus on how students broke apart the given shapes and determined unknown dimensions in order to calculate the volumes.

Here are some questions for discussion:

• “How did you approach solving this problem?” (First, I broke the barn and silo down into separate shapes.)
• “What shapes did you break the barn and silo down into?” (Barn: rectangular prism and triangular prism; Silo: cylinder and hemisphere.)
• “How did you find any unknown dimensions?” (We didn't know the height of the triangular prism. I knew the total height of the barn was 7 meters, so I subtracted the height of the walls, 5 meters, to get 2 meters.)