Today I wanted to create a dome that would have an 8mm inside
diameter and a 9mm outside diameter (.5mm sheet). I had absolutely no
idea where to start to estimate the right size disk for starting -
and hence I have two very sore spots on my fingers from so much
sanding down! DOH! are there any formulas for this? or good rules of
thumb even?

I am mathematically challenged so if this is obvious I do apologize

There may be other ways to figure it but what you want is the
surface area of a hemisphere converted to the surface area of a
circle. The surface area of a sphere is 4 x pi x the radius of the
sphere squared. So a hemisphere is one half of that or 2 x pi x r
squared. This is the area of the disc you need to cut to make the
sphere. So you then figure the square root of the area of the
hemisphere divided by pi. This is the radius of the disc you need.

I am sure that many have formulas for disc sizes, When I was a
goldsmith apprentice I was taught by the workshop silversmiths to
add the diameter of the dome to the height as this gives a rough disc
size, but sometimes if you are hammering a larger disc to a dome some
metals will stretch more than others giving you a larger dome. Then
it is down to some filing.

Mathematically if you measure the inside diameter of the required
dome and multiply this figure by 3.142 ( which is PI to three
decimal places) then divide by 2, this will give the exact finished
interior curve measurement of the domeās interior, but this
measurement does not allow for any metal expansion. In my experience
a hammered doming punch will expand the metal very slightly, where as
a slow press will not.

I wanted to create a dome that would have an 8mm inside diameter
and a 9mm outside diameter (.5mm sheet). I had absolutely no idea
where to start to estimate the right size disk for starting - and
hence I have two very sore spots on my fingers from so much sanding
down! DOH! are there any formulas for this? or good rules of thumb
even?

Surface area of a sphere is 4 Pi R squared. For a 1/2 sphere divide
by 2. I donāt remember why but a little voice inside tells me to use
an R which about 2/3s of the way through the metal thickness. Of
course at your size you will be daping, and the metal will thin and
stretch. But even if the math is not perfect due to distortions of
space/time, the area of the hemisphere and disk should be pretty
close.

Some things are not possible to explain but using mathematics, so
you are going to have to persevere.

You need to start with metal disk of the area, which when domed
should become of required size. There are few principals involved:

Deformation - when metal is deformed, the outer layer stretches,
and the inner compresses; the middle layer is unchanged. That is why
a pipe is as strong as a rod. Therefore all calculations are done
using middle layer. In you case it is 7.5mm.

Geometry - a dome is 1/2 of a sphere, so calculating area of a
sphere and dividing by 2 would yield the result.

A(surface area of a sphere) = Pi( constant = 3.14 ) * D^2(diameter
of a sphere squared) = A = 3.14 * ( 7.5 * 7.5) = 3.14 * 56.25 =
176.625 = 176.6 mm^2 ( one decimal place is good enough precision )
176.6 / 2 = 88.3 mm^2 which is area of the starting disk.

However, when we work with metal, metal stretches. By how much, will
depend on individual technique. For beginners it may be necessary 20%
correction. As technique improves, the amount of correction
decreases. Using 20%, the area of starting disk becomes 70.6 mm^2.

Area of a disk = Pi*r^2, so r (radius of starting disk) = square
root( A / Pi ) = square root ( 70.6 / 3.14 ) = 4.74 mm So diameter of
starting disk is 4.74 * 2 = 9.5 mm approximately.

Using 10% correction - sqrt((88.3 * 0.9)/3.14)*2 = 10 mm

Since amount of deformation depends on the thickness of the metal (
the same technique will result in more metal stretching depending on
thickness ) a formula incorporating this dependence is useful.

D disk = (D(outside) - thickness of metal) * Pi * (1 - Correction) /2

D disk = ( ( 8 - 0.5 ) * 3.14 * ( 1 - 0.1) ) / 2 = 10. 6 mm

So the range is from 9.5 to 10.6. You would need to experiment to
find out what is your number is. Important to remember individual
technique can significantly influence any formula.

Nothing works like real life. I dapped a 12.7mm disc until it had a
2:1 ratio of diameter to height(assuming a true hemisphere). Finished
diameter was 9.9. Interpolating, that means your disc should start
out about 28% larger diameter than what you want for finished
diameter of the dome.

Factor in gauge and distortion. Real life being what it is you
probably donāt want to hand cut an oddball size disc when a tool
punched disc might be close enough. Do you really NEED a true
hemisphere or will close enough be close enough? If you can get away
with a shallower dome you could start with the next available
undersize disc and dap until you have your desired diameter, the
height being whatever it is.

For math challenged people, Rupert Finegold and William Seitz in
their book Silversmithing, provide a useful method for visually
determining the disc size needed for a certain bead size. They
present a method for determining bowl sizes for hollowware but it is
equally useable for determining bead sizes.

I have included this method in my book, Doming Silver Beads, along
with discussion and other methods (including formulas). The
explanatory graphic on the Finegold-Sietz method is from my book and
is as follows:

In addition, the estimated size of a bead based on the size of a
24-gauge (0.511 mm) disc is presented below.

It is based on using standard Sterling silver. Gold, Argentium
Sterling silver and copper all have more stretch, so the sizes listed
below will be slightly too large. As others have stated, it also
depends on how hard you strike the metal. The best way to approach
this is to start with a disc size and see what size bead you get,
mark it down, and go to the next size, etc.

Disc Diameter Bead Diameter

3/8" 7.3 mm
7/16" 8.4 mm
1/2" 9.5 mm
9/16" 10.7 mm
5/8" 11.8 mm
11/16" 12.9 mm
3/4" 14.0 mm
13/16" 15.2 mm
7/8" 16.3 mm
1" 18.6 mm

I find that all the math in the world will not predict exact
finished pieces. Experiment will and I have made good scrap in order
to obtain the domes I want. I would counsel to use copper as a silver
substitute or silver for a gold substitute and experiment with the
sizes and dome shapes and still expect to have adjustments when you
start in with the metal you want to work in.

Just as the differences between the architect and the engineer or the
engineer and the construction worker, any building or hand made item
will differ from the drawing or my thoughts and until I actually make
the piece I will not know the process. I usually build in or have to
accept those differences when designing, even after multiple decades
of metal work. I see it as the differences between my thoughts and
reality.

I often am suspicious of formulas. In my studio practice, I try to
use an analog or stand in for the real thing. If I want a specific
dome in 18ga sterling, Iāll grab some 18ga coper, brass whateverā
even wax sheet-- make an educated guess at disc size, dome it and
record the resultā¦

I have a few ready to mount pieces, bezels sitting down nice and
tight. Yup the teacher had me do a formula to measure the bezel, she
watched my every move. Not one stone fit. Now need some lapidary
work done or new stones cut to size.

Wrapping a 1/4 inch wide strip of Quilterās Tape from the local
Craft store works just fine.

I guess the x .9 is to account for the metal stretching.

If you want to use .9, preliminary shape must be done with wooden
punches, and when switching to steel, change hammer to rubber-faced.
The distortion will be limited only to stretching the outer layer.

This is one of many calculators. A simple search on any type of
calculator will usually yield results. This Google search was on
āSphere calculatorā. Today it is the best and simplest way to start.

http://www.csgnetwork.com/circlecalc.html This is one of many
calculators. A simple search on any type of calculator will usually
yield results. This Google search was on "Sphere calculator".

That appears to calculate a sphere with the same radius as the
circle, and not the radius you would need to bend the surface into
the sphere.