### Packing a triangular piece of glass?

A glass prism is to be shipped in a cylindrical tube that has an inside diameter of d inches. Given that the cross section of the prism is an equilateral triangle with side of length p and the prism fits snugly into the tube, write p as a function of d.

FeralFeline

on November 30th, -0001

This is a great question… requiring some small amount of trig (the law of sins) and some basic logic.

The answer first= f(d) = 1.1547p

The hard part is going to be explaining this without drawing it.

If you draw the triangle in the circle, then draw a line from a point of the triangle, bisecting the base, and then touching the edge of the circle. In essence, you have just drawn the diameter of the circle.

Now do that with the other two points as well. You are left with a point in the middle of the triangle.

Take a look at 1/3 of the big triangle now… where p is the base of the smaller triangle and the radius is the other two sides of the triangle.

This creates a triangle with angles of 30, 30 and 120. (Hopefully you can see that because again, it’s hard without the drawing here.)

Finally, solve for one of the sides of r, by using the law of sins. Let p = 2 for this situation:

sin 120 / 2 = sin 30 / r

Solving for r = 2 (.50) / .866

r = 1.1547

hence, d = 2.3094

and p = 2

Since the geometry is proportional as angles don’t change, we can solve for d by divinding p into d to get the appropriate ratio. hence:

d = 1.1547p