-1

I have this problem and I understand most of it, but am having trouble deriving the formula for the side of each rhombus. I think I just can't visualize it well. Can anyone draw a picture that would help me find the side length of the rhombus?

"Let S be the intersection of two cylinders with radius r that intersect at angle theta. Find the volume of S as a function of r and theta.

enter image description here

Jennifer Durost
  • 1

1 Answers1

1

Consider one of the cylinders, and take a plane parallel to its axis and at distance $0 \le d \le r$ from it.
The intersection of the plane with the cylinder will be two parallel lines, which will be at distance $2\sqrt{r^2-d^2}$ from each other. Can you visualize that ?

Then given two cylinders, whose axes intersect, the two axes will lay on a plane which cuts each cylinder in half. Any plane parallel to that and at distance $d$ from it, will cut each cylinder in a couple of lines as said before.

Those two couples of lines, inclined by the angle $\theta$ are making the rhombus.

G Cab
  • 35,272