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The Circular Stairs

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The height at the top of the 2nd floor requires that a person must be elevated 11 feet within one 360° turn.
The circular stairwell was divided into 4 quadrants. The steps start in Quadrant 1. Working up and around the staircase, Quadrant 4 is located just before the 2nd floor landing.

The first 2 steps in Quadrant 1 are regular rectangular steps;    then the circular arc of steps run the rest of the way to the 2nd floor.
  
To reach the height of 11 feet in exactly 18 steps, a step rise of 7.334 inches is required.
 
18 x 7.334" = 132" = 11 ft.
The next consideration is headroom.    When a circular staircase turns around on itself, the 2nd floor lies over the top of the lower steps.    There must be enough headroom so a person doesn't hit his head on the lower decking of the 2nd floor while climbing or descending the lower steps.
Fortunately, the distance between the height of the 3rd step and the bottom of the 2nd floor decking, turned out to be a very acceptable    8 ft., 2 inches.

( 7.334" x 3) + 12" = 34.002"

then

132" - 34.002" = 98" = 8 feet 2 inches
Next I had to figure the actual lay of the 18 steps.    Steps #1 and #2 are in Quadrant 1 - so is step #18, which is the 2nd floor.

Quadrants 2 thru 4 form 3/4 of a circle or 270°.
Steps #3 thru #17 (i.e., a total of 15 steps) are located in these 3 quadrants.
  
Thus 15 steps have to be positioned within 270° of a circle.    Doing the division shows that each step occupies 18° arc.
  
270° / 15 = 18° arc depth for each step.
   
The width of a step from the center of the center pole to the inside of the banister is 3.66 feet (assuming the banister is 4 inches wide).

Using the formula for the circumference of a circle, it can be figured that each step will be 1.1498 feet (or 1 ft, 1.8 inch) deep at the outside banister - a reasonable depth for a circular staircase step.

(Pi = P = 3.1415)    (Radius = R = 3.66 ft.)
C = 2PR = 22.9958 ft.

(3/4)C = 17.2468 ft.

17.25 / 15 = 1.1498 ft.
This is the staircase when drawn (approximately) to the above scale.    You see it this way when entering the stairs from the hall to go upstairs.
This is the way it appears when descending from the 2nd floor.
And last, but not least ... the space beneath the steps, in quadrant 4, becomes the hall coat closet by the front door.



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