How to Construct the Wheel of Theodorus

To construct your own Wheel of Theodorus using Euclidean tools, follow the steps described and illustrated below.

1.  Construct right triangle ABC, with base = 1 unit and height = 1 unit.

2.  Using hypotenuse AB as the base of the next triangle, construct a perpendicular to line AB through point A.

3.  Mark off 1 unit as the height of right triangle DBA.

4.  Construct triangle DBA.

5.  Continue using the hypotenuse of the newly constructed right triangle as the base of the next triangle iteration. The height should continue to be 1 unit.

6.  Continue constructing your “Wheel of Theodorus” until the final picture is pleasing to the eye.

7.  What is the length of the hypotenuse of the final right triangle? Explain or show how you know using the Pythagorean theorem.


About Ms. Miles

I've taught mathematics for over 22 years. I love teaching math. Interests outside of teaching include learning and doing math, playing the piano, singing in a worship team, origami, kayaking, bicycling, swimming, collecting sea glass and shells.
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15 Responses to How to Construct the Wheel of Theodorus

  1. Courtney says:

    Very helpful! Thank you!

  2. Sam says:

    Why did it stop at root 17?

  3. ankur says:

    thanks Ms. Miles. I have a test of this tomorrow in my tutions

  4. Alex says:

    Thank you! I used this to hep me with a project!

  5. marisol prieto says:

    How would I explain this to someone who has never heard of it? I am doing a project in my math class and I have to explain it in a way that children from k-8th grade can understand. Please help! Thank you!

  6. Ms. Miles says:

    Hello Marisol, I might begin my project with a description of what the wheel of Theodorus is, in simple terms:

    “The Wheel of Theodorus is an interesting design which has a nautilus spiral outline, but which is constructed using a series of right triangles. This design illustrates the connection between mathematics and nature. In addition, it shows the connection between art and math. The history of the design originates with Theodorus, a Greek mathematician of the 5th century BC and a pupil of Pythagoras … (research this further). The Wheel of Theodorus begins with a simple right isosceles triangle of base and height 1 unit. Let’s call this figure Stage 0. To construct Stage 1, build a second right triangle of height 1 unit whose base is the hypotenuse of the triangle constructed in Stage 0. (consider defining the terms right triangle, base, height, hypotenuse). The design continues in this way, with each newly constructed right triangle having a height of 1 unit and a base of the previous stage’s hypotenuse. Over time, a spiral will begin to emerge. This spiral also has significance in mathematics, as it can be enclosed by a rectangle that is known as the “Golden Rectangle”.

    An interesting feature of the Wheel of Theodorus is that although it is a fully unified figure from Stage n to Stage n+1, each new stage reveals a hypotenuse which is either rational or irrational in length. This can be shown using the Pythagorean Theorem.

    Hope that helps you get started.
    Ms. Miles

  7. Aniket says:

    I have done my project with help of internet

  8. Rohan.R says:

    Ms.Miles i have a project tomorrow and i just did the wheel but it stopped at 16 but i saw your website and i did it it definitely helped me thanks a lot.

  9. Unknown says:

    mrs miles Thanks for the help

  10. pinky says:

    thanks a lot

  11. Poornima devanga says:

    Thank you mam it’s so helpfull

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