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2 π Problems

First, some review: if a line segment is rotated 360 degrees around one of its endpoints, a circle is traced out.

Problem One: Rotate a scalene triangle 360 degrees about a fixed vertex of the right angle, and the endpoints of the two legs a and b will trace out two concentric circles. The interior area contained between the two circles is called an annulus. First, why is the mathematician's (Enrico Fermi) diagram incorrect for this problem? Second, show that the area of the annulus is the π (b2-a2), where a < b.

Problem Two: Rotate a scalene triangle 360 degrees about the point of intersection of the hypotenuse c and the longer leg b, and the endpoints of these two sides again will trace out two concentric circles. Show that the area of this annulus is the π (a2), where a < b.

Related questions or explorations for both problem situations:

  • What happens if you start with an isosceles right triangle?
  • What happens if you start with a 30-60-90 right triangle?
  • What happens if you start with an isosceles non-right triangle?
  • What happens if your start with a random triangle?

 

Source: Carl Sparano's Letter to Editor, Mathematics Teacher, December 1993


Hint: Draw a picture for each situation and find the area of the annulus via the areas of the two circles.

 


Solution Commentary:

Problem One: Area of annulus = π b2 - π a2 = ?.

Problem Two: Area of annulus = π c2 - π b2 = ?.