Geometry – Day 61 – Circles, Sectors, Segments, and Annuli

[N.B. These are taking for-ev-er to crank out, for whatever reason, so I’m changing the format a bit. If anybody misses the minute by minute breakdown, I’ll bring it back. I’m guessing it was only there for formality’s sake, though.]

Come for:

  1. The area of circles and all their chopped-up, little parts.

Stay for:

  1. Famous Idiots in History
  2. The Stop Sign Project


  1. None


Slide Deck

  1. honey. Snapple Real Facts. They know what (-x,y), (x,-y) and (-x,-y) look like. These are (y,x). What would (-y,x) look like? (There are at least two answers for b, incidentally.)
  2. The Slate article:
    The US mint page:
  3. We all acknowledge that paper bills are the ideal heist. If you HAD to steal any coin, you’d want to steal dollar coins for value. But if you consider WEIGHT in the mix, are dollar coins still your best bet? After you decide, how much would 45,000 pounds of each be worth? What does a low rate mean?
  4. We’ve known area of a circle for awhile. I don’t really want to talk about the formula.
  5. Do you see the parallelogram
  6. We’ll make it clearer.
  7. Now do you see the parallelogram? Alright, what’s the height.
  8. The base is a little trickier discussion.
  9. So there it is.
  10. A basic example.
  11. Working backwards from the answer.
  12. 15 is a great question on what happens to the area of a circle when you double the radius.
  13. Show the three figures, give “doughnut” as a more realistic name. What does the area of each depend on?
  14. Homework: Determine the area of a stop sign in square inches.

Notes & Revisions:

  1. The Algebra proved more difficult for my kids than I anticipated.
  2. Notice that dimes, quarters, and half-dollars are all worth the same relative to their weight. That isn’t an accident.

I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.


  1. In the opener: a) is a reflection. b) could be translation, and it could be a reflection, or it could be translation followed by rotation (many of them), or it could be reflection followed by rotation, or……
    More realistic name: usually people refer to a torus as a doughnut. Both are rather questionable. Torus is two dimensional (surface), annulus is one dimensional, doughnut is 3 dimensional.

  2. First — Love the circle area dissection. It hints at limits, which is nice. I feel like I just saw this demo for the first time very recently, maybe in the NCTM newsletter?

    Also, do you have issues with red/green colorblind kids doing your opener?

  3. E, I just can’t seem to get this translation/transformation thing right.

    Mr. C, Yeah, yeah, I’m such a fan of limits, both in mathematics and as a logical exercise. I wish I could think of more gentle ways to introduce them. Especially, I would love to derive area of a circle by sending the # of sides of a regular polygon to infinity. I’d lose half the class inside the first slide, but it might be worth it for the sake of the second half.

  4. Oh, and no one’s outed themselves as r/g colorblind, but, playing the odds, I should probably work on my color selection. Crud.

  5. I note a hint of sarcasm in your response. I’ve noticed before that you seem bit defensive about these lessons. Somebody said that they could understand it since you clearly put a lot of work into them. I agreed, but also thought that since you are so concerned with teaching that you would also be concerned with what you teach. Does it seem like a petty peave to you, this issue of translation vs transformation?

    How do you respond to a student who says “I just can’t seem to get this translation/transformation thing right”? Do you say “Oh, well, it’s ok, no worries”? Or is it not ok?