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

Calculus for Geometry Students

In Geometry, we’re beginning a chapter on areas of polygons, and the first lesson is area of a rectangle. Pretty exciting, huh? My students are mostly ninth-graders, with a few tenth-graders, and I thought they might enjoy seeing how the area of a rectangle is used to estimate the area under a curve, i.e. a Riemann Sum.

I used a Geogebra activity created by Alex Kasantsidis to demonstrate a simple Riemann sum. We discussed how the sum of the rectangles can either overestimate or underestimate the area under the parabola, and how we can get a better approximation of the area by increasing the number of rectangles used.


Then, I had my students work through an activity (you can download it here) to estimate the area under the curve y = 12 – x^2 for x = -1 to x = 3 using eight rectangles. After averaging the left-hand and right-hand sums, they came up with 38.5. The actual area is 38.66…, so with only eight rectangles they achieved very good results!

What my students enjoyed even more, though, was the satisfaction of learning calculus-level mathematics. Hopefully, this activity allayed some of the apprehension they might have when they hear the word “calculus”.


Visualizing Punctuation

At, Adam J. Calhoun has come up with a fascinating way to look at famous literary works: take away all the words and only look at the punctuation. Don’t believe punctuation matters that much? Compare Cormac McCarthy’s Blood Meridian (on the left, below) to William Faulkner’s Absalom, Absalom (on the right):


I shared his findings with all the English teachers at my school, and they were really interested in the information one can glean from this analysis.

Check out Calhoun’s post to see some beautiful “heatmaps” of famous novels. They look like Rothko paintings.

Algebra + Area = A Nice Proof

In my Geometry classes, it is time to work with right triangles. Rather than simply present my students with the Pythagorean Theorem, I decided to have them prove it without them knowing that’s what they were doing.

I based the following activity on a proof I found at Alexander Bogomolny’s very useful site, Cut-The-Knot.

A Proof

  1. What is the total area of the four triangles below? (They are all congruent)

 4 rt. triangles



 2. If the triangles are rearranged into the square pattern below,

Pythagorean Proof Square

(a) In terms of a and b, what is the area of the “hole” in the middle?



(b) What is the overall area of the square pattern?



3. How can you use the results of parts 1 and 2 to prove a famous theorem?


I like this proof, because it’s very visual as well as algebra-based. The most challenging part for my students was figuring out the area of the square “hole”. Many assumed the side of the square was 1/2 of a, which is not correct. The satisfaction and excitement they had as they saw the pieces of the proof come together were well worth the effort, though! They were all familiar with the Pythagorean Theorem, but this was the first time any of them had actually proved it.

Click here for a pdf of the worksheet and here for the solution.