Geometry – Day 59 – Review & Regular Polygon Area

Why We’re Here:

  1. We’re going to review area.
  2. We’re going to determine the area of regular polygons.


  1. Remote Control Car

The Breakdown

  1. Opener + Review (15 minutes)
  2. Race Car Math (30 minutes)
  3. Break (5 minutes)
  4. Show and Tell (1.5 minutes)
  5. Polygon Notes (20 minutes)
  6. Polygon Classwork (30 minutes)


Slide Deck

  1. 1680. Boston Globe. Ask them to convert this to hours per day. Lots of ways to solve a and b. Entertain all of them.
  2. Other secondary rules: have the driver pick one place to steer and stay there; thumbs off the control sticks when the judge calls time, or he awards five bonus seconds to the other team. Be quiet as the solution is explained or the other team is awarded five bonus seconds.
  3. All work and no play …
  4. #8 is fun as they all wonder how to solve for area without the height. If no one in your entry-level Geometry class answers “30 square millimeters,” you’re the winner.
  5. In one class, I started by deriving a formula (basically flipping this slide with the next one) and then going for a specific case. The next class I pulled a switch and everyone was, foundationally, a little better off. Ask here: what shape can you split the regular polygon into?
  6. New vocabulary. Now develop the formula at the board using these variables. Talk about p = ns
  7. Ask them to ID the building. After they answer, the Pentagon, the teacher is encouraged to ask obliviously, … and what kind shape is the Pentagon?
  8. They built twice as many bathrooms as necessary due to segregation. The center was thought to be the most likely site of a nuclear attack and called ground zero. The rings are named A-E from the center outwards.

Notes & Revisions:

  1. None
  2. Several values corrected per David’s suggestion. Thanks, David.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. In #8, I don’t see how you don’t have enough information (and if you don’t, then how did you find the area?). They surely know Pythagorean theorem, no? If not how to prove it, then at least what it says. Then you look for a different altitude, and find that the area is 2sqrt(221) approx 29.93

  2. Pythagorean Theorem happens early next week. And it’s tough to tell here without actually looking at the QuickTime or Keynote that “3 mm” builds in after they recognize they don’t know enough with just the three sides.

  3. I thought you teach high school. In which case you may not have taught Pythagorean theorem yet, but usually kids have seen it in middle school. Regardless, I know that the 3 appeared after the “Not enough information”, but one of the things I was trying to point out is that 3mm is not correct (and that 15, 15 and 4 is enough information, if you know Pythagorean thm). I think it would be good for them if you revisited this problem once you teach Pythagora, that way they can see that now they can solve the problem they previously could not.

  4. This is a very good lesson, but I noticed a lot of diagrams with impossible measurements. I think that it would be better to use measurements that are consistent with the Pythagorean Theorem, even though you haven’t covered it yet.

    The triangle in 1(c) (from the opener) is inconsistent, since the area (according to the diagram) is 50 sqrt(6), not 100. I would suggest changing the height to 15 and the area to 300.

    The parallelogram in Race Car #2 is impossible. The length of the slanting side must be greater than the height, but your lengths are 12 cm and 19 cm respectively. Changing the length of the slanting side to 24 cm would fix this.

    In Race Car #8, the actual height is about 3.964 cm. (It is very interesting that the height is so close to 4 cm.) I might change the side lengths to 100 mm, 100 mm, 43 mm, and the height to 42 mm.

    Race Car #11 is a very nice problem. I wonder if anyone noticed that the missing area is 35*39/15.

    The apothem of a regular heptagon with side 10 is approximately 10.4. (exact value is 5*cot(pi/7))

    I love the Pentagon problem!

  5. Yikes. You two auditors are gonna make me regret this resolution of mine.

    This may reflect our different populations here, E, but, particularly in a review exercise, it’d be a bad play for me to say, “You guys know how to do this. It’s the Pythagorean Theorem. You did it in eighth grade.”

    We’re reviewing with that problem and high schoolers (excepting an honors population, I suppose) require a certain focus to their activities. This also means I can’t just break down and say, “Let’s prove this,” when we happen upon some theorem that we could prove.

    David, out of respect for your fact-checking, I’m going to update this with your accurate numbers. I appreciate your efforts and I welcome future nitpickings, if you’re up to it. These lessons would only be stronger with more accurate measurements.

    That said, accurate measurement ranks pretty low priority-wise with lessons that soak up an average four hours apiece. I try to keep things generally to scale and I respect the Triangle Inequality, which my kids know. I just don’t know which of my Geometry kids will call me out on my inaccurate trigonometry.

  6. Dan,

    Of all people out there I would have thought that you’d welcome constructive critisicm and corrections. I would imagine that David would agree with me that our goal was not for you to regret putting your lessons here for the world to view. It is impressive that you are willing to reveal your work this much.

    …but, particularly in a review exercise, it’d be a bad play for me to say, “You guys know how to do this. It’s the Pythagorean Theorem. You did it in eighth grade.”

    Why? They do know it. Why not use it? “I haven’t thought it yet” could be an answer, but it seems to me that teaching them the proof is not going to happen, so what more can you teach them than they already know?

    You should not update your lesson out of respect for David’s fact-checking. You should update it because they are incorrect. Or, the numbers are not possible given the picture and assuming we are in Euclidean geometry. Is it nitpicking to point that out? I don’t think so.

    Also, I don’t believe David was telling you that you should measure your figures’ sides in your presentations. He was telling you that the numbers you gave can not appear as sides of the triangles. Let’s look at the example he mentioned:

    The parallelogram in Race Car #2 is impossible. The length of the slanting side must be greater than the height, but your lengths are 12 cm and 19 cm respectively. Changing the length of the slanting side to 24 cm would fix this.

    He is saying that the right angled triangle in that picture formed by a side of parallelogram and the altitude can not have those lengths: hypothenuse can not be longer than the leg. One way to fix it is to change the number 12 to something bigger than 19. Also, this is not inaccurate trigonometry, it is inaccurate geometry. And just because your kids will not call you on that, should that be a reason not to do it correctly?

    Look, I am trying to have a conversation about things that I think are important: mathematics. My goal was/is not to attack you. It is to help improve something that you clearly care about and put a lot of time and effort into. If you don’t welcome that, and since this is your space, I will stop.

  7. Dan,

    I hope that my nitpicking will be seen as a positive reflection on your work. Your geometry lessons are of very high quality, and it is very generous of you to make them freely available. If the lessons were merely average, then I would not have bothered to check the numbers.

    Also, I appreciate e’s clarification that I was talking about logically possible measurements, and that I was not suggesting that the figures should be drawn to scale. In fact, it’s probably better if the figures are not to scale.

  8. No no no. The term nitpicking sounds so negative, but I was serious in my appreciation. Keep it up, you two. This is making things a lot better in my classroom.

  9. Oh right, but, E, the whole Pythagorean Theorem deal. I’m having a hard time explaining why, pedagogically, it would be a bad idea for me to inject Pythagoras into a review of area. It wouldn’t be altogether untowards for me to ask the class, in a spirit of total academic curiosity, “do you guys know how you might solve for the height here?”

    But in terms of telling everyone, “Alright use Pythagorean Theorem here,” when twelve kids are still struggling with triangle area and 24 kids have forgotten it, would’ve been a poor play. In my classroom. With my population.

    I’m not sure else to describe or qualify the conundrum.