Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Thursday, November 4, 2010

Special Right Triangles

 You know the ones...

Convenient, nice to know going into trig, time-saving...especially for the SATs and GREs. It's a little hard to believe just how much the College Board hearts it some special right triangles.

But still, hard to motivate.

Enter...the humble dollar bill. It's a rectangle. What sort of rectangle? Not all rectangles are the same shape, of course. Some are squares! Their sides are in a 1:1 ratio. Some are square-like. Some are long and skinny. A long, skinny one's sides might be more like 10:1 or even more severe.

What about our paper currency in the US? All denominations are the same size and shape. This is not  true in every country. But ours are all this very familiar rectangle.

What ratio do you think its sides are in? If you use two bills to measure, to see how many short sides make up the long side, you can see that it's skinnier than 2:1 (If it were 2:1, it would make a square when you folded it in half, and it doesn't. Credit for that observation: Rachel B., class of '13), but not quite 5:2 (or 2.5:1, if you prefer). (Cue annoying kid shouting "TWO AND A THIRD! IT'S TWO AND A THIRD" employing a technique commonly known as "proof by intimidation.")


Now we do a little folding. I handed out photocopies of dollar bills. They were black and white, and one-sided, so I don't think it was a felony.

It's an isosceles trapezoid!

It's a rhombus!

It's an equilateral triangle!
At this point, you can also gently unfold it and coax it into a tetrahedron.

Unfold it all the way, and behold...

Don't see it? Look at the creases.

What does this mean? In my classroom it means I can launch the derivation of side relationships in 30-60-90 triangle with a tiny amount of Davinci-code-type intrigue which is totally worth navigating the murky waters of incommensurate lengths. But in a nutshell, it means our familiar dollar bill is made of a rectangle in a supremely weird ratio of $4:\sqrt{3}$

and come on, that was no accident. I bet the Freemasons were involved.


  1. And look at how the pyramid shows up in the triangle after you fold it! It's a conspiracy!

  2. imaginative visual proof, this. rattling off trig vlues of standrad angles becomes second-nature later on and as an engineering student this really appealed to me - the actual physical reality.
    I'm off to see what i can do with our currency now!

  3. Yet again, I am floored by your teaching genius.

    And so once more, I find myself forced to wonder why, for the good of the Republic, we do not all simply lay down our red pens and agree unconditionally to make you our supreme leader and mathematical overlord.

    BTW, seriously, I am stealing this idea.

  4. As much as I love a good conspiracy theory, especially one tongue-in-cheek, it is more likely based on a few limiting factors - standard paper sizes and/or paper making equipment.

    U.S. notes have not always been the size they are now. Since the paper for U.S. notes is custom made, it is still possible there was some behind-the-scenes influence....

  5. I am honored to have been in the presence of this monumental discovery.

  6. You have done it again! This is a super cool way to get students to just enjoy the fun of learning.

  7. Your top diagram here is wrong, 2a and root 3a are around the wrong way.

  8. The ratio doesn't seem so weird when you express it as "16:3"

  9. Well, except that squaring both sides of a ratio doesn't leave you with the same ratio...16/3 doesn't equal 4/root(3).

    Other than that, Justyn, you'd be right.

  10. @thomasdav I'm not sure what to say except, no they're not. Think about how you can make a 30-60-90 by dropping an altitude in an equilateral triangle. Or, think about how the hypotenuse has to be the longest side.

    @Justyn squaring two numbers doesn't keep them in the same ratio. Since you are multiplying them each by different quantities.

  11. @Sean I showed the kids the dollar bill shirt you made, too!

  12. Washington was born in 1732. 1.732 is an extremely close approximation of √ 3.

  13. Pro tip: Math teacher bloggers obsessively check their work on math posts before posting.

    @Kate: How many times larger would a dollar bill t-shirt need to be to fit on a person?

  14. George Washington was born in 1.732... and there are FOUR SYLLABLES IN "GEORGE WASHINGTON!!!"

  15. okay. i like it. but how is
    "put one equilateral triangle
    next to another with their
    bases along a line; inscribe
    the whole thing in a rectangle"
    anything like "supremely weird"?
    square roots are just *built into*
    plane geometry (and our allegedly
    intuitive notions of "distance").
    one gets used to it after a spell.
    (n.b. the whole eye-in-the-pyramid
    thing is indeed quite creepy;
    another story though...)

  16. If you know you're starting with two eq triangles and drawing a rectangle around it, of course it's no surprise at all. But if you're wondering about the rectangle and then notice the triangle thing, it's surprising. At least, I was surprised, and a little delighted. I guess one person's weird is another person's ho-hum.

  17. Look at the top of the outlined triangle. The E is directly in the center on the top. This E stands for EYE and is the THIRD letter 'the'.

  18. How incredibly cool! I just found this blog, and have enjoyed your passion for teaching mathematics so much. I don't know if you might be interested but many math teachers seems to need a better way of drawing and writing mathematics. I know I did so my husband wrote a program that embeds in Word that has helped me tremendously. It's free and can be downloaded at Let me know if you like it!

  19. I found this site today and thought of you and this assignment if you are interested. lets you make your own personalized money.

  20. I am not sure where the best place to post this on your blog would be so I am choosing your most currect trig entry. I love to read all of the blogs and find yours to be one of the best concerning trig so I'm hoping you can help. I teach trig at a high school where our AP Calculus teacher believes in the right triangle method for teaching how to find the sine of an angle, such as 210 degrees. Although we absolutely use the unit circle a great deal for many things (visualizing radians compared to degrees, finding sin and cos of quadrantal angles, knowing the signs of the quadrants, seeing how the graphs are formed), when it comes to finding the sine of something like 210 degrees, the students are absolutely not allowed to do that on a unit circle. They need to know that it has the same sine as a 30 degree angle, other than the positive or negative, and then they need to know the sine of 30 degrees by use of a 30-60-90 triangle or memorization. She strongly believes that we are doing the kids a disservice when we go through how a unit circle is made with triangles once but then allow them to memorize their way around with various tricks and mnemonics. The students are actually forbidden to sketch the unit circle for a case like this as she does not believe it leads to understanding, just memorization. Therefore, my students know all about the unit circle except for actually memorizing the points for the common angles. Do you have any thoughts on this? I've searched for a discussion on this topic and can't seem to find a good one!

  21. Hi Kathy - I'm not sure how you could understand that 210 in standard position gets you a 30 degree reference angle without sketching/understanding the unit circle. Also, I don't see how it is possible for a teacher to forbid a student to reason a certain way. But maybe I am missing something.

  22. Perhaps I didn't explain it well then. They definitely use the unit circle to visualize where 210 degrees is and we do use the unit circle often for determining the reference angle or the radian measure. However, for actually finding the sine of an angle, we are supposed to teach them how to find the sine of 30 degrees based on a triangle rather than having them memorize the coordinates that go around the circle. She tells them right away that they should not have to count around the circle to find the sine or cosine of an angle, rather, they should understand what sine is and be able to instantly know it from a triangle. I will also add that she is an amazing teacher and her AP scores are phenomenal. We have one trig teacher at our school who uses the unit circle and it frustrates the calculus teachers that the kids need to draw a circle or count around it to find the sine of 210 degrees!

  23. Hm, well, it kind of seems like micro-managing at a level that I don't engage in. However, I understand the frustration with imperfect recall of memorized facts... but I don't get the sense that my students memorize coordinates on a unit circle, either. They tend to figure out which quadrant, draw the triangle, and use the sides of special right triangles. Do you have a preferred method, Kathy? Do you think it's better for teachers in the same school to standardize?

  24. I definitely believe the kids need to have a thorough understanding of the unit circle in order to fully understand the graphs and values of the trig functions. However, when it comes to actually finding the sine of 210 degrees, I am undecided. I do see a lot of kids who need to count their way around the circle to find that value and I'm not convinced that is best. I would much rather they know that it is the same (other than the sign)as the sine of 30 degrees and then find that by knowing it is opposite the short leg and is, therefore, 1/2. I did find an example or two of studies that compared students' retention and understanding of the trig functions given both methods and they seemed to say that the right triangle approach is better in the long run. In answer to your second question, I think it is great for the kids if they have been taught to come at a problem a variety of ways so, no, I don't think we all need to teach exactly the same way. That said, I also believe that while I am teaching trig, I need to be preparing my kids to handle the calculus course they will be taking. If I can teach some of the concepts similarly to the way they will be learning them in calculus, I think it will benefit them when they get to that course. Honestly, I have done so much research on the two methods of teaching the trig functions. I know that I am not a fan of "tricks" for memorization...I'm one of those teachers that asks the kids "why" all the time and wants them to know why they are doing something rather than memorizing a trick.

  25. I just found your blog (actually, I just found the whole edublogsphere in general), and I love this activity. I'm using it tomorrow.

    I've been going two and a half years now in this job without thinking to start looking online for people sharing stuff like this. OOPSIE.


  26. Hi, Malcolm. Welcome to your new addiction. :)

  27. Addiction - no kidding. My whole weekend just vanished. This is fantastic.

  28. Great teaching strategy! Every time I read one of your posts I'm impressed at how you come up with one of these. Engaging kids on a visual level is something that I want to employ! Thank you!

  29. Very cool. Sadly, the ratio is different for Canadian money. Of course, I will happily accept any of your donations.

  30. Awesome intro to Special Right Triangles!! Just planned to use this dolla bill inquiry activity with my 8th grade pre-algebra kids next week. Love your blog!!

  31. I have been teaching for about 20 years and have only now discovered blogs. WOW - I am redoing a lot of my lessons - Thank you! I think my students will definitely like this as an intro. Just making sure I am not missing something - Once they unfold the bill, how do I get them to find the ratio?

  32. Hi Sheryl! Welcome! If you haven't already, make sure you see this page.

    How you get them to find the ratio is kind of on you. :-) It depends on what class it is, what the kids already know, how comfortable they are with open-ended investigation...

    Personally, I think I'd do some version of
    - draw an equilateral triangle on the board
    - draw an altitude
    - ask them to write down everything we know for sure
    - ask them to articulate what it is we want to find
    - wait until someone remembers that they know the pythagorean theorem
    - troubleshoot their algebra mistakes

    Good luck! Let me know how it goes.

  33. I've basically done the equilateral triangle thing exactly as you describe it, minus the intro with the dollar bill - Definitely makes things more interesting. I have not checked out "this page" yet, but I will. I think I need a couple of more hours in my day now that I've discovered this new world.

  34. you know when people comment on youtube videos and say something like "who's here still listening in february 2019"?? THIS IS ME DOING THAT NOW. i did this in class yesterday and the kids LOVED it! why were they so excited when i handed out paper money? i have no idea. but it was a perfect intro to 30-60-90 triangles and super interesting in and of itself. THANK YOU KATE


Hi! I will have to approve this before it shows up. Cuz yo those spammers are crafty like ice is cold.