# Was Pythagoras wrong? - related heresies

- Author:
- Judah L Schwartz

*In that applet we established that there is a sequence of paths that come closer and closer the to the hypotenuse of an isosceles right triangle. Each of the paths in the sequence has the same length; i.e., the sum of the lengths of the two perpendicular legs of the triangle. In an isosceles right triangle with hypotenuse of length*

**Please look carefully at the applet “Was Pythagoras Wrong?” before exploring this applet.***, the length of each of the other legs is*

**a****Thus the sum of the lengths of the two perpendicular legs is**

*a**and the perimeter of the isosceles right triangle is*

**a***The perimeter of the shape formed in this applet, i.e., the region bounded by the two perpendicular legs (green) and the “sawtooth hypotenuse” (blue) is*

**a***. Challenge: What is the area enclosed by the shape formed? What do you think it means for a curve to be smooth? If the distance between two points that are normally thought of as being a distance*

**a***apart can be as much as*

**a***apart, what are the implications for other polygons that have such “sawtooth” edges?*

**a**