# Circle Stacking

1. Jan 21, 2009

### Mentallic

Given 3 circles each of radius r, stacked up into a triangular pyramid shape, find the height of the entire structure. This might be expressed more clearly in a graphical form:

http://img206.imageshack.us/img206/3646/stackcircleseb0.png [Broken]
http://g.imageshack.us/img206/stackcircleseb0.png/1/ [Broken]

I haven't been able to answer this question for years! Any suggestions as to how to find the height h would be appreciated.

Last edited by a moderator: May 3, 2017
2. Jan 21, 2009

### daniel_i_l

Draw a triangle connecting the centers of the circles. Calculate the height of the triangle using trivial trigonometry. Can you proceed from here?

3. Jan 21, 2009

### Mentallic

Ahh it would form an equilateral triangle, giving me the angles I have searched long and hard for to find!

$$h=\frac{6+\sqrt{3}}{2}r$$

That one line:
all it took to build that bridge I've been longing for. Thankyou very much, I greatly appreciate that one line of help

4. Jan 21, 2009

### KnowPhysics

I too worked out some and i got this:
h=2r+sqrt((2r*2r)-(r*r));

5. Jan 21, 2009

### D H

Staff Emeritus
You are correct that the triangle is an equilateral triangle. Your derived height is however incorrect. If you arrived at this result we can help find the error in your reasoning.

6. Jan 21, 2009

### Mentallic

Seems I was too excited and skipped a whole lot of rational thinking.
I knew it would've been a stupid mistake, and while I don't know what I did wrong yesterday (no point in trying to find out), I have the correct answer now.

$$(2+\sqrt{3})r$$

which is equivalent to KnowPhysics' after a bit of manipulation:

$$2r+\sqrt{(2r)(2r)-(r)(r)}$$

$$2r+\sqrt{4r^2-r^2$$

$$2r+\sqrt{3r^2}$$

$$2r+\sqrt{3}r$$

Thanks