Man Floyd, you sure know how to tax an old man's brain cells and prod them into several years of uncharted territory. 

First, you are correct, it will be longer than that.  In order to accommodate the radius and the elevation change that only stands to reason.

If you start a standard stair you first calculate your height from floor to floor.  Then, you must know what your available code allowed rise and span distances are.  You take a measurement that is comfortable to the customer (older short people would prefer less rise than younger tall people) and within the code and divide it into the rise.  Say: 140" rise divided by 7" per tread = 20 risers/treads.  Now this will depend upon how you do the final step/landing area but for simple that means you a bottom landing, 19 steps/treads, and then one last up onto your floor/landing.

Then, you take your span and multiply that, usually 11", by the number of treads, in this case 19, to get 209" or 17'-5".  So, you will be 17'-5" out from your floor/landing edge.  That doesn't include the bottom 'imaginary' step on your bottom floor (remember, you actually have 20 steps, you can look at the bottom floor coming down or the top floor going up as number 20) so you have an actual distance from edge to edge of 18'-4".  With those measurements you can calculate your running diagonal using Pythagorean: square root of sum of a squared and b squared.  260.77" diagonal.

Now, that was a round about way of saying, if that measurement is longer than the horizontal run on standard stairs, why would anyone think that the length and/or radius of your spherical stairway would be the same as if measured on the flat?  And the radius will make yours longer than the standard would be.  So, for same height as in this illustration, instead of 260" yours may be about 300".

Now, for your radius stairway...I will have to find my calculations book I used when we used to build these in the shop.  I don't need these formula when inspecting. 

Remember, code will tell you what your minimum span and maximum span on a spherical tread can be.  You have some things to figure out.  You can use what I already said to figure out how many treads you will need based upon code numbers. 

Brent