# For a sphere, why is the derivative of the volume the equation for the surface area?

Yeah, so the title of the post says it all. I am teaching a standard calculus course, and I wanted my kids to see why this beautiful thing holds true.

It’s not a coincidence. And in fact, a circle also has a nice property: the derivative of the area of a circle ($A(r)=\pi r^2$) gives you the circumference of the circle ($C(r)=2\pi r$). So yesterday I decided I wanted to come up with a short investigation that at least exposes my kids to this idea.

After working for around 90 minutes this morning, I ended up with a packet, and these things on my desk which I’m going to use for illustrations (blocks, dumdums, and tape):

UPDATE: I was shopping yesterday and found these gems. YAAAS!

I’ll post the packet I whipped up below. It goes through the standard argument, so in that way it’s nothing special. But in the past when I taught the course, I used to just kinda stand up at the board and give a 5 minute explanation. But I wasn’t sure who was really grokking it, and I was doing too much handwaving.

The big picture trajectory:

*At the start of class, but way before doing this activity, I’m going to have kids recall what a derivative is graphically (the slope of a tangent line), and then how we approximated it before we used limits (the slope between two points close to each other). And from that, I’m going to remind kids of the formal definition of the derivative:

* I may also start class with this problem, suggested to me on twitter by Joey Kelly (@joeykelly89). It’s a classic problem that was featured on xkcd, but oh so unintuitive and surprising!:

*Way later in class, I will transition to this activity. The first idea is to get kids to see the connection between the volume of a sphere and the surface area of a sphere. And then again for the area of a circle and the circumference of a circle.

*Then I try to get kids to understand what’s going on with the sphere first… followed by the circle.

*Then I show kids the “better explained” explanation. Why? Because at this point, kids are spending a lot of time thinking about the algebra, and I’m afraid they might have lost the bigger picture. The algebra focuses on one “shell” of the sphere, or one “ring” of the circle. But how does it all fit together? [@calcdave sent me this video, which I’d seen before but forgot about, which has the same argument… this is where the licorice wheels above come into play.]

*Finally, I problematize what they’ve learned. I have them mistakenly make a conjecture that the derivative of the volume of a cube is going to be the surface area of the cube, and the derivative of the area of a square is going to be the perimeter of the square. But quickly kids will see that isn’t quite true. So they have to tease out what’s happening.

My solutions:

I haven’t taught this yet. So it could be a complete disaster. I don’t have a sense of timing. I don’t know how much of this is me and how much will be them. I am just hoping tomorrow isn’t a disaster! Fingers crossed!

1. For your “Take it further” you could do
equilateral triangle and tetrahedron
and
see if you get something similar, with common factors, as with square as 2x2s and cube as 2x3sxs

1. Very exciting. What a great idea. Thank you!

2. 1) When I tried to do something similar-ish earlier this year, the “divided by h” part of the derivative was the hardest to explain/motivate. I think the flattened dum dum image leading to SA*h is really clever, and I wonder how that worked.

2) The “take it further” part is very cool. I wonder if you could also tie this to the rectangle representation of the product rule.

1. 1) It actually worked super well for 80% of the kids. I am not sure about 20%. I wish I had started with the circle/ring *first* because they all seemed to grok that well.

2) Ahhh, we did that visualization. Yes! I need to think about how I’d do that without having kids get lost. But maybe throwing up the slide where we did that and having them look at that along with what they generated with the square might be good? (We didn’t have time to get to the cube/square, sadly.)

3. The perimeter is the derivative of the area of the square, if you remember to paramterize by the distance from the center to the middle of the edge (8x and 4x^2).