SOLUTION: I want to build a dome with a cir. of 60' and the top cut off to provide a 3' dia. hole (cir.=9.24') in the top. I would like to know the circumference of 7 additional rings besid

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Question 106435: I want to build a dome with a cir. of 60' and the top cut off to provide a 3' dia. hole (cir.=9.24') in the top. I would like to know the circumference of 7 additional rings besides the 2 mentioned of 60' and 9.24'. Each positioned equadistance up the outside of the sphere. The idea to build a geodesic dome with triangular suports between each outside ring.
Found 2 solutions by Fombitz, solver91311:
Answer by Fombitz(32388) (Show Source):
You can put this solution on YOUR website!

A circumference of 60' gives a radius of

so then at C=60 ft, then
R=9.549 feet
The radius at the top is R=1.5 feet.
The y dimension between the 60 ft circumference (R=9.549) ring (Point 1) and the R=1.5 ft ring (Point 2) can be calculated by finding the y dimensions of each and subtracting.
You can do this since both points lie on the circle with radius R=9.549.
For point 1,
x=-9.549, y=0
For point 2,
x=-1.5 and

The difference in y is

This is broken up by 7 additional points into 8 equidistant spacings of 9.431/8=1.179 feet.
We can then calculate the corresponding x values that go with the y values.
The x values are then the radii of the 7 additional rings and we can measure the circumference.
Let's start.
Point 1 is at y=0, each other y will be a multiple of .

The corresponding x's are calculated using

with the results

These are then the radii for each of the rings and the circumference is equal to,

Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!
Draw a cross-section of your dome, i.e a semicircle. Drop a perpendicular from the point of intersection of your top hole and the dome. Also construct a radius from the center of the sphere (center of the circle described on the ground by the placement of the dome) to that same point of intersection. You will now have constructed a right triangle with one side equal to the radius of your hole at the top and the hypotenuse equal to the radius of the dome.
Call the top hole radius Rs and the dome radius Rd.

First convert your dome circumference into a radius:

or 9.55 feet.

Now the angle between a dome radius at the ground and the dome radius to the intersection of the hole at the top of the dome (and therefore the size of the subtended arc) is given by:

or in your case,

(about 81 degrees)

Just because we will need to refer to it later, let's call this angle A1.

Since you are going to add 7 rings, you need to divide this arc into 8 equal pieces.
1.413 radians/8 = 0.1766 radians is the measurement of the arc between any two of the rings forming your dome, and we will call this angle A2.

Starting from the top, counting the first of the 7 intermediate rings as number 1 and numbering consecutively as you go down the dome, the radius (Rn) of any ring would be given by:

Where n = the ring number as defined above.

Once you have calculated all 7 radii, use

to determine the circumferences

For example, to find the radius of ring #3:

and the circumference would be
or 38.08 feet
You can do the rest of the arithmetic.