# Calculating Pyramid Height from 3 Circles

• Mentallic
In summary, the conversation discusses finding the height of a triangular pyramid made of three circles stacked on top of each other. The solution involves drawing a triangle connecting the centers of the circles and using trigonometry to calculate the height. The final result is h=(2+\sqrt{3})r or 2r+\sqrt{3}r.
Mentallic
Homework Helper
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
http://g.imageshack.us/img206/stackcircleseb0.png/1/

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

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:
daniel_i_l said:
Draw a triangle connecting the centers of the circles.
all it took to build that bridge I've been longing for. Thankyou very much, I greatly appreciate that one line of help

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

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

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

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.

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

## 1. How do you calculate the height of a pyramid using 3 circles?

To calculate the height of a pyramid using 3 circles, you can follow these steps:

• Measure the diameter of each circle.
• Find the average diameter by adding the three diameters and dividing by 3.
• Divide the average diameter by 2 to get the radius.
• Use the formula h = √(r^2 - (r/2)^2) to calculate the height, where r is the radius.

## 2. Why do we need to use 3 circles to calculate the pyramid height?

Three circles are used because they represent the base of the pyramid, which is a triangle with three sides. By using three circles, we can calculate the average diameter and radius, which are crucial in determining the height of the pyramid. If we only used one or two circles, the calculation would be inaccurate.

## 3. Is this method of calculating pyramid height accurate?

Yes, this method is accurate as long as the measurements of the circles are precise and the formula is applied correctly. However, this method may not be suitable for pyramids with irregular bases or for pyramids with a circular base.

## 4. Can this method be used for any type of pyramid?

No, this method is most suitable for pyramids with a triangular base. For other types of pyramids, such as square or rectangular based pyramids, a different method of calculation may be required.

## 5. Are there any other methods for calculating pyramid height?

Yes, there are other methods for calculating pyramid height, such as using trigonometry or geometry. These methods may be more suitable for pyramids with irregular bases or for more complex pyramid structures.

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