When I created the Spiral of Theodorus it required me to use a protractor to measure for the angles and the different sizes of the triangles. You use the protractor to make sure all your triangles measure to be exactly 90 degrees. Each triangle is supposed to be 1 inch apart. For the math component of this project I used the Pythagorean theorem to check that my triangles were accurate.
In order to create the spiral(aka the rams horn) You are going to need a protractor, because without it you are not going to be able to do the math portion. You can start off by putting the dot anywhere on the blank piece of paper that you have. Once you have your dot you want to create a line that is an inch long. Then create another line, that is also an inch, where it would create a 90 degree angle. Then connect the 2 ends of those lines in order to create your right triangle. From there you are going to use the hypotenuse from that triangle as your base for the next triangle. You will then repeat the process. The spiral of Theodorus is based off of the pythagorean theorem. The equation a^2+b^2=c^2. When you simplify the answer for c^2 by square rooting it, that will be your hypotenuse for that triangle. When you see the hypotenuse for all the triangles you will see a slight increase in your numbers as you keep adding triangles. You will get to a point where there will be no more room for any right triangles. That is when you know you have finished. If you're feeling creative you can make your own design with the spiral, such as I did by making it the horn of a ram.
First I used a protractor to make a dot in the middle where the spiral would surround. I used the protractor to line up the triangles to the point at a 90 degree angle every time, I used the Pythagorean theorem to check that my triangles were correct. I made each of the triangles 1 inch toward the 90 degree angle which would then connect it to the next triangle.