I've been thinking of creating different kinds of buildings using guidelines similar to Conjuring the Mystic Tower and was wondering how I should go about doing this.

What I have a mind is a dome using 10,000 cubic paces of stone but I'm not sure how I should go about calculating the dimensions. Any ideas?

Volume of a sphere is: V=(4Pir)^3/3
I assume a dome is a half-sphere, so half that.
You know the volume, so you isolate r
Assume a sphere with a volume of 20.000 cubic paces then

So r= (cube root(3V/4 Pi)

Insert 20.000 cubic paces as V, I get a radius of 16,8 paces.
So a dome 16,8 paces high and a circular footprint with a diameter of 33,6 paces

No, don't do this, as it gives a solid hemisphere, which is not what you're after if you're making something to live or work in. Also, the exponent is in the wrong place: 2 pi r^3 /3. The exponent got fixed in the calculation, though.

While you could take the difference in two such volumes, the much faster way is a thin-shell approximation: V=2 pi r^2 t, where r is the radius and t is the thickness. Set your thickness and solve for the radius. Or just set both to find the volume required.

If you're making this like the tower, though, you'll also want a significant foundation. If you double the volume, leaving half the stone for a solid foundation, that will give you a foundation with double the shell's thickness.

I completely agree. You only need the volume of the walls, not the empty space inbetween.

I can't think of a way to calculate that...maybe start by assuming a set thickness of the walls/roof. This may have too many variables to just solve. Maybe one just needs to guess at some sizes and adjust up or down until it fits.

Googling "Volume of a spherical shell" gives me: V = 4/3 • π • (r³ - (r-t)³)
r is outer radius
t is thickness

I'd rather have inner radius, because that defines the available space, but what the hey.
Should we assume a dome thickness of 2 paces? I'd hate to have to guess about the real-world engineering of this, but It's a magical creation. Maybe there are examples of cast concrete domes?

So that's
20.000 = 4/3 * π/ * (r³ - (r-2)³)
(3/4 * 20.000)/π = r³ - (r-2)³
... and then what?

I gave a simpler formula above. As for an actually necessary thickness, I decided it was better to ignore the physics of that. Just choose a somewhat reasonable-seeming thickness, like 1 pace. Toss that and the radius into the formula. That's why I did the similarly simple doubling suggestion for the foundation.

If I were making better suggestions, I would give a shape other than a hemisphere, where you would get more usable space and where it would support itself better. The usable space issue is that the outer parts of the space will have a very short ceiling if it's actually hemispherical, so that will effectively remove living area; you want the walls to rise faster. It happens that you can make it support itself better with walls that rise faster, too.

The Pantheon in rome (the classic unsupported spherical dome) is about 6.5 metres thick at the base, but only about 1.2 metres by the hole at the top. This is presumably because the neutral surface (a parabola) has to stay inside the stone at all times to make sure it is in compression, and a sphere doesn;t quite work for this.

This is what I was referencing above in regard to shape, but you don't need wider blocks at the bottom to fit such a curve inside it. Rather, as things will look much more linear and much steeper, that reasoning would tend to make it thicker higher up. Rather, what happens with the load is that the top doesn't need to take much, while the bottom needs to handle much more. So you generally build thicker at the bottom to handle the bigger load. This is reversed in hanging, so a chain hung between two spots horizontally across from each other takes the most load at the top and the least at the bottom.

(Edit: Callen gentle reminded me than 1 and 10 aren't the same thing on the following post)

Also the Pantheon is made of bricks, and giving them internal cohesion forces you to work with more of them.

If you want to get an idea of the width you are going to need, you should look for concrete domes instead. I suspect that 1 meter width will already be over-dimensioned and would work just fine even not having a parabolic section (but also that anything less would make a medieval magus feel paranoid about it).

About the foundations, half of the stone placed there also seem to be a bit overkill, but good for the sake of simplifying the math; considering 1 meter width, the 10000 cubic paces turn into a hemisphere of an area of 5000 m^2, so 5000 = 2 π r^2, thus r = 28 m, and on the ground you would end with an area of half the dome's area, 2500 m^2.

A wall in a hemispherical dome is considered "thin shelled" if the wall thickness is 10% (or less) of the radius. This 10% proportion is considered safe for a full hemisphere which goes 90 degrees from crest to base.

That was wrong too: The Pantheon is made of coffered, unreinforced concrete. To date, the biggest unreinforced concrete structure ever made, actually (partly. I guess, because from one century ago all concrete is reinforced).

Note to self: next time check Wikipedia before posting. And don't miss 0's. Apparently sometimes they do count for something, despite what the advertising said.