Pythagorean Spiral

Materials : Chart paper, pencil, ruler, set squares, coloured pencils or markers.

Step 1: Mark a point P on the top left corner of the chart paper as shown in Fig (i).

Step 2: Form a triangle to the right of the point P as shown in Fig (i). It is a right isosceles triangle with the equal sides as 10 cm.

Step 3: Calculate the hypotenuse of the right triangle using Pythagorean theorem.

c2 = a2 + b2, where 'c' is the hypotenuse
c2 = 102 + 102 = 200
c = √(200) = 10√2

Step 4: Form a new right-angled triangle with 'c' as the base and another side of 10 cm containing the 90° (as shown in Fig (ii)).

Step 5: Calculate the new hypotenuse(d)

d2 = (10√2)2 + 102 = 300
d = √(300) = 10√3

Step 6: Repeat steps 4 & 5 with 'd' as base. The new hypotenuse 'e' will be equal to 10√4. Refer Fig (iii).

Step 7: Continue this process 10 more times.

[As you will see, the hypotenuses will be 10√2, 10√3, 10√4 (or 20), 10√5, 10√6 ..... 10√14]

Step 8: Did you get a spiral pattern as in Fig (iv) ? Colour it creatively !