Original Question:
I am building a rather large flower pot out of western red
cedar for a palm tree. The bottom is 3 ft. across and will have
12 sides measuring 29 inches high each. The sides will lean
out at 15 degrees. The width of sides at the bottom measure
10 & 1/2 inches. Is there a mathematical formula to calculate
the width of the sides at the top? Also, can the miter angle
be pre-determined. I already know that the bevel will be 15
degrees. Any help would be greatly appreciated.
Solution:
 Click
on this calculator link
to open a calculator window.
Diagram 1 represents a side view of the planter. The
angle formed by the intersection of lines G and H is the 15
degrees of lean you require. 
- G & J depth of planter
- R is radius of bottom
- U is the radius at the top of the planter
- S is the width of the side at the bottom
- T is the width of the side at the top
Diagram 2. The open center represents the bottom of
the planter. I have shown only a few sections of the 12 sides
(dodecagon). The yellow portion represents the top of the pllnater.
Since the sides lean 15 degrees the radius of the top dodecagon
will be larger than the bottom.
 From
the information you gave in your question:
R would be 18 inches (half of the 3ft width of the bottom).
H is 29 inches (height of the sides)
S is the length of the bottom side. It should be 2*18*sin(15)
= 9.32 inches (this is shorter than the 10.5 inches you mention
in your question). If 10.5 inches is the critical measurement
then the width would be twice R where R = (10.5/ 2) * (1 / (sin15)
= 20.28 or width of 40.57 inches.
To calculate the width of the sides at the top of the planter
(T) we use the formula of:
T = 2 * (R + H * sin 15) * sin 15
T = 2 * (18 + (29 * .2588)) * .2588 = 13.2
Diagram 3.
 Diagram3.
This diagram shows the sides of the planter.
The bottom dimension is S and the top is T.
The equation below the diagram shows how to find the angle between
J and H.
For the dimensions in this case:
tan x = ((13.2 - 9.32) / 2) / 29
tan x = 0..0669 (use ArcTan or Tan -1 to find x)
x = 3.82 degrees
y = 86.17 degrees
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