This activity clarifies the distinction between volume and surface area and illustrates that two polyhedra can have the same volume but different surface areas.

Students build figures using two sets of 8 cubes and determine their volumes and surface areas. Because all of the designs are made of the same number of cubes, they all have the same volume. Students then examine all of the designs and discuss what distinguishes figures with smaller surface areas from those with greater ones.

As students work, monitor the range of surface areas for the figures that students built.

This activity works best when each student has access to snap cubes. If physical cubes are not available, consider using the digital version of the activity. In the digital version, students use an applet that has 16 cubes with which to build the two figures.

Ask all students to display their designs and their sticky notes and give students a couple of minutes to circulate and view one another's work.

Then, ask previously identified students to arrange their designs in the order of their surface area, from least to greatest, and display their designs for all to see. Record the information about the designs in a table, in the same sequence. Display the table for all to see. Here is an example.

Give students a minute to examine the designs and the information in the table. Then, discuss the following questions:

• “What do all of the shapes have in common?” (They have the same volume.)
• “Why are all the volumes the same?” (Volume measures the number of unit cubes that can be packed into a figure. All the designs are built using the same number of cubes.)
• “Why do some shapes have larger surface areas than others? What do shapes with larger surface areas look like?” (The cubes are more spread out and have more of their faces exposed.)
• “What about those with smaller surface areas?” (They are more compact and have more of their faces hidden or shared with another cube.)
• “Is it possible to build a shape with a different volume? How?” (Yes, but it would involve using fewer or more cubes.)

If students have trouble visualizing how surface area changes when the design changes, demonstrate the following:

• Make a cube made of 8 smaller cubes. Point to one cube and ask how many of its faces are exposed (3). 
• Pop that cube off and move it to another place.
• Point out that, in the “hole” left by the cube that was moved, 3 previously interior faces now contribute to the surface area. At the same time, the relocated cube now has 5 faces exposed.

Previously, students studied shapes with the same volume but different surface areas. Here they see that it is also possible for shapes to have the same surface area but different volumes. Students think about how the appearance of these shapes might compare visually.

Students are given the side lengths of three rectangular prisms and asked to find the surface area and the volume of each. Some students can visualize these, but others may need to draw nets, sketch the figures on isometric grid paper, or build physical prisms. Prepare cubes for students to use. Each of the three prisms can be built with 15 or fewer cubes, but 40 cubes are needed to build all three simultaneously. (If the cubes are not centimeter cubes, ask students to treat them as if the edge length of each cube was 1 cm.)

As students work, look out for errors in students’ calculations in the first question, which will affect the observations they make in the second question. Select a few students who notice that the volumes of the prisms are all different but the surface areas are the same. 

The goal of this discussion is to highlight volume and surface area as distinct measurements of a polyhedron. 

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written statement about volume and surface area by correcting errors, clarifying meaning, and adding details. 

• Display an image of Prisms A, B, and C for all to see.

  

• Display this first draft:
  Prism C has a greater volume than do Prisms A and B, so Prism C also has a greater surface area than the other two prisms.
• Ask, “What parts of this statement are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, and then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

If needed, consider referring to the filing cabinet activity in an earlier lesson to help students conceptualize the idea of figures with different volumes having the same surface area. The number of square sticky notes needed to cover all of the faces of the filing cabinet was its surface area. If we use all of those square notes (no more, no less) to completely cover (without overlapping sticky notes) a cabinet that has a different volume, we can say that the two pieces of furniture have the same surface area and different volumes.