# Volume of a dome

1. Jul 25, 2006

### laminatedevildoll

We have a dome in our lab, and my advisor asked me to calculate the its volumne and make it into a sphere or spherical. This'll be used for some thermal calculations I think. What should I measure and what equations would I need?
I'd appreciate anyone's feedback.

2. Jul 25, 2006

### Werg22

Try to establish the relation between the base and height and use an integral.

Edit: Here is more detail.

First and foremost, you have to realize that in any 3d figure where the area of the base is a function of the height, we can use an integral to calculate the exact volume of the figure. If you don't already, then there is no other way.

Imagine your dome upside down (this is easier). At height 0, the base area is 0 and at maximal height, the base's area is maximal. Let's denote the maximal height by H and the maximal area of the base B. Now you need to find a function such as:

b = f(h)

and

B = f(H)

and

0 = f(0)

Where b is the area of the base and h is the height. Now to get the volume you use a simple integral:

$$\int_0^H \ f(h) dh$$

Of course, if you don't already know these concepts, this will sound like total gebberish to you.

Last edited: Jul 26, 2006
3. Aug 4, 2006

### Robokapp

I'd picture it seen rotated so it's highest top is at (0,0) and it's lying on its side so the x-axis is going trough its middle. Like a cup tipped over...with the x-axis piercing its center. Then describe it by a formula. If it's a dome, I'm guessing a square root formula could work.

Then you integrate using the disk method.

Pi * Integral of f(x) * f(x) from zero to the furthest point...or the height of the dome.

I had in my 12th grade about finding volumes of different flower vases and other containers and it was very interesting I thought.

Edit: It's Calculus AB chapter "Application to integrals" knowledge.