Question

how to make square root spiral

Answer 1

How to create a spiral of sequential square roots:
1) Construct an isosceles right triangle ABC with side 1 inch. (You can use the corner of an index card to make sure you are constructing a perfect right angle.) The hypotenuse AC will then by equal to the square root of 2.

2) Construct line at point C perpendicular to segment AC. Construct segment CD on this new line, equal in length to segment AB. Construct segment CD on this new line, equal in length to segment AB. Connect points A and C. Construct a line at C perpendicular to AC. Construct segment CD equal in length to segment AB or BC. Then construct segment AD. AD will then be equal to the square root of 3 (use the Pythagorean Theorem to calculate this length).You now have 2 right triangles, with one leg of the second triangle formed by the hypotenuse of the first triangle. You now have two right triangles, as shown below:

3) Construct a line at C perpendicular to AC. Construct segment CD equal in length to segment AB or BC. Then construct segment AD. AD will then be equal to the square root of 3 (use the Pythagorean Theorem to calculate this length). Construct a line at D perpendicular to AD. Construct segment CD equal in length to segment AB or BC. Then construct segment AD. AD will then be equal to the square root of 3 (use the Pythagorean Theorem to calculate this length).You now have two right triangles, as shown below:

4) Continue this process until you have 11 right triangles as shown below. If you have used a compass and straightedge, trace the segments onto a clean sheet of paper, without the construction marks.


4) Use your imagination and create a picture out of this spiral that shows something you may see that has a spiral shape in nature – a shell, a snake, the profile of a human head…think big!
You must turn in both the calculations on the POW sheet, your work to create the spiral, and the clean, imagined spiral.

Grading rubric
• Include a title for your picture.
• On the front of your picture, include your signature and the date.
• Label all triangle legs and hypotenuses with appropriate lengths.
• Conjoin each new right triangle with the hypotenuse of the previous right triangle.
• Make sure your project is neat.
• Use color unless you mean to emphasize contrast by using black and white.
• Write your labels using radicals unless they can be simplified to rational numbers. For example, you might label a hypotenuse 9 = 3.
• Connect all of your hypotenuses to the same central point.
• Attach the POW paper to your art containing your calculations to find lengths of segments (using the Pythagorean theorem) for your first 8 triangles.