Illustrating Volumes of Solids with Known Cross Sections

My posts are few and far between these days.  I hope that changes soon, but for now, it’s just a post here and there when I can.  And today I can!  Let me share some fun Desmos-ing with you!


In this post:

  • A new Desmos feature: how to use labels!  
  • Attending to the clarity and usability of graphs.  
  • Fun cross-sectional solids graphs to play with!!  What could be better?

It’s no secret with my colleagues that I love Desmos.  For real. SO. MUCH.  And I love it when people ask me to help them look for Desmos graphs or activities for particular topics.

Which brings us to today. One of the calculus teachers I work with was asking me about whether I knew of any good Desmos graphs of rotational solids.  I sent him some awesome graphs made by Geoff Patterson.  Graphs that use such advanced math that my jaw drops with awe.  Seriously.  If you haven’t given his blog a look, do yourself a favor and check it out.  Pronto. Here’s a link:


This picture is of one of Geoff Patterson’s creations. Image can be found at

But what I didn’t know was that this colleague also wanted an awesome Desmos graph illustrating solids made with known cross sections.  Apparently, he had seen stuff on Geogebra, but nothing was fitting the bill.

So I took note of his wish list.  It included these must-haves:

  1. editable functions
  2. editable bounds
  3. varying numbers of visible cross-sections, from very few to nearly filled-in
  4. rectangular cross sections with constant height
  5. rectangular cross sections with proportional height
  6. semi-circle cross sections
  7. equilateral triangle cross sections
  8. isosceles right triangle cross sections

I told him it would certainly be doable in just an evening, but that if he didn’t need it right away, I may not tackle it immediately.  He left school for the day, and assured me that he didn’t need it soon.  As in, he just wanted it for next year.  So I put it off, procrastinator that I am.

Just kidding!  As if I could wait to get started on such a fun-sounding graphing challenge!  I love tackling this sort of problem, so I worked on it, despite having other tasks on my to-do list.

So what have I been learning recently that I applied in making these graphs?  Let me share.

Labels!  This is a new feature as of this week, and it was awesome to be able to label f(x) and g(x) on the graph.

Pro-tip #1: To get a label, make a point, then click “show label” and write whatever you want the label to show.  



Pro-tip #2: Want just the words and not a point? Click on the circle to the left of the point to hide it!  The label will still show up.


I made labels for f(x), g(x), the lower bound of the region, and the upper bound.

Also, with regards to clarity and usability, I’ve begun making brief annotations in my expressions list to label what’s being done where, and what certain variables I define mean.  I haven’t always done this, and have regretted it later when I come back to a graph and want to change something, but can’t remember which variable represented what, and therefore have to think through all the math again.

annotation 1.png

I’m going to urge you to begin doing this, too, if you haven’t already!  It’s easy, and makes such a difference!

Pro-tip #3: Yes, it’s annoying to have to use a mouse and click to “add note.”  So don’t! Did you know that you can just type ” and Desmos will add a note instead of an expression?  Ah.  Keyboard shortcuts.  Gotta love ’em.

And here are the resulting graphs!

Rectangular Cross-Sections of Constant Height:


Rectangular Cross Sections with Proportional Height:


Semi-Circular Cross Sections:


Equilateral/Isosceles Triangular Cross Sections:


I’m still working on the right triangle cross-sections.  I’ll post when it’s done.


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