Modeling Solids of Known Cross-Sections

One of the hardest type of problem for calculus students to understand is calculating the volume of solids of known cross-sections. It’s hard, because they have difficulty visualizing it.

Last year, I went to the regional NCTM conference here in Nashville, TN, and one of the sessions I attended addressed this exact issue. Nina Chung Otterson was the presenter, and she teaches at The Hotchkiss School in Connecticut. She has her students cut cross-sections of different shapes and apply them to a base area enclosed by two parabolas, y = x^2 – 3 and y = 3 – x^2.

Here’s what the base area looks like, courtesy of’s online function grapher:

Base area

In her session, Nina Otterson provided templates that fit the given base area for different shapes: semicircles, squares, and equilateral triangles. I had my students cut the square templates diagonally for isosceles right triangles, and horizontally for rectangles. Students use the templates to cut out a cross-section that fits down the middle of the base area, and six others on each side.

Here are my students in action, cutting out the cross-sections:

Cutting Pieces 3 Cutting Pieces 1 Cutting Pieces 2

Here they are, taping the cross-sections onto the base area:

Building Solids 1 Building Solids 2

And here are the finished models:

Squares Equilateral Triangles Isosceles Right Triangles Rectangles Semicircles

Once they understood that the thickness of the paper was dx, it was very easy to set up the integrals to calculate the volumes of their models. I’ve never had students grasp the idea behind this type of volume as quickly and as easily as this group did. Building a model using actual cross-sections made all the difference!

When I do this activity next year, I think I’ll glue the base area to foamboard, and have students insert the cross-sections into slits cut into the foamboard. That way, they will stand up straighter and stay evenly spaced.

You can download the templates provided by Nina Otterson here.

Update: I used spray adhesive to glue the base area to some foamboard and cut slits in it with an Xacto knife. Then I carefully slid each cross-section into its appropriate slit. It worked great! Now each cross-section stands nicely spaced and vertical.

Final Version


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