The purpose of this activity is to give students the opportunity to reflect on their own understanding and apply that understanding to solve problems involving volumes, the Pythagorean Theorem, and trigonometry. Students rely on their knowledge of the solids and how the volume of each is calculated to determine which have all the information needed given and which require additional steps, such as adding auxiliary lines (MP7).

Look for students who choose a right prism with which they can straightforwardly compute , students who choose the oblique cylinder that doesn’t involve trigonometry, and students who choose a prism that involves trigonometry.

Students may not be sure how to visualize the height of the oblique cylinder with the 40-degree angle. Remind them that they can add auxiliary lines to the diagram.

The purpose of this discussion is for students to share strategies for finding volumes of prisms and cylinders. Ask students:

• “For which solids did you think the volume would be easy to calculate? Why?”
• “For which solids did you think the volume would be hard to calculate? Why?”
• “For the solids with missing measurements, how did you decide which dimensions you needed to calculate?”
• “What were some of the dimensions that were missing from the figures?”

The purpose of this activity is for students to apply what they have learned about prisms and cross-sections to a context.

The goal of this discussion is to connect Cavalieri’s Principle to the tower volume calculation. Ask students to imagine a building that is identical to the Cayan Tower, except that the floors are rotated so that they line up with each other, forming a right prism. Ask students:

• “How are the floors of the building like cross-sections? How are they different?” (They are horizontal slices like cross-sections, but cross-sections have no thickness, whereas the floors do have thickness.)
• “Even though the floors aren’t exactly cross-sections, we can use reasoning involving cross-sections to compare the volumes of the original building and our imaginary rotated building. How do the volumes compare and why?” (The original building and the one with rotations have equal-area “cross-sections” (or rather, equal-volume floors) at all heights, so their volumes are the same.)
• “How does this reasoning help us find the volume of the tower?” (The expression  gives the volume of the imaginary rotated building because it is a prism. This volume is equal to the volume of the tower.)