Just double-checking my math.

• Flag A has proportions 5:3. Width is <w> and height is <h>.
  
• There is a white star with five points, all spaces 72^o apart.. All convex angles are 36^o. All concave angles are 72^o.
  
• The circumscribed circle of the star's outer points shall have a diameter (1/10)w.
  
• The circumscribed radius of the star's outer points is <r>, and the inscribed radius is <r₂>,
  
• To coordinates of the star's center are <3r, 10r>.
  

To roughly describe that to people not plotting this out with me, that means that there is a star one tenth the length of the flag in the upper-left-hand corner. Now for a description of what I want to accomplish with that star, before I get to the math involved in accomplishing that.

Surrounding the white star shall be thirteen stripes, each of equal thickness. The stripes shall conform to the shape of the star. Some of the star-shaped outlines will not be fully visible, but will be implied on account of the parts that appear on the banner. The thirteenth striped will only have one of its concave corners visible -- in the bottom-right-hand corner. The main star will be tilted in such a way so that this corner and the corner of the flag match up perfectly, and that there is neither the beginning of a fourteenth stripe, nor a thirteenth stripe that is not as thick as the others.

Now for the math on how we get there. I will start by defining r₂ with respects to r.

• The outer corner and nearest inside corner of the star form a triangle.
  
• The lengths of two of its sides are r and r₂.
  
• The angle between them is half of 72^_o.
  
• The angle opposite of r₂ is half of 36^o.
  
• The angle opposite of r is 180o minus the other two angles.
  
• 180-(72+36)/2=126
  
• I can determine r₂ via the Law of Sines.
  
• r₂=r(sin18/sin126)
  

Now, if I were to draft a version of this flag by creating multiple layers on Photoshop, one for each concentric stripe which, in its own layer, would be a full star, by how much would I have to upscale the star in each layer? How much larger must each star be in order for the inside corner of one of them to be exactly the distance from their shared center so as to match the distance of the flag's bottom-right-hand corner to the same point?

• The distance from the star's center to the bottom-right-hand corner can be determined through the quadratic formula.
  
• The coordinates of the bottom right-hand corner are <20r, 0>.
  
• The triangle formed between this point and the star's center has legs <3r-20r, 10r-0>.
  
• h²=[(-17r)²+(10r)²]^1/2
  
• h=r389^1/2
  

This isn't quite enough. You would think that you simply take this number and divide it by 13, but that will not come out as intended. The distance covered by the stripes does not cover the hypotenuse of these two points, but rather the hypotenuse minus whatever space is already covered by the original star.

• h₂=r389^1/2-r(sin18/sin126)
  
• h₂=r(389^1/2-sin18/sin126)
  
• h₂=r(389^1/2sin126-sin18)/sin126
  

In order to find the thickness of each stripe, measured from each of their inner radii to the inner radii of the next stripe down (or in the case of the first stripe, the distance from its inner radii to the inner radii of the original star), simply divide by 13. Stripe thickness shall be given by the letter

• Stripe thickness is <t>.
  
• t=r(389^1/2sin126-sin18)/(13sin126)
  

Now I must find out the exact distance from each inscribed stripe radius from their shared center with respect to r.

• Inscribed radius is given as <r_2.n>, where n is the number of the stripe in relation to its proximity to the central star.
  
• r_2.n=r₂+nt
  
• r_2.n=r(sin18/sin126)+nr(389^1/2sin126-sin18)/(13sin126)
  
• r_2.n=r[(sin18/sin126)+n(389^1/2sin126-sin18)/(13sin126)]
  
• r_2.n=r(13sin18+n389^1/2sin126-nsin18)/(13sin126)
  

In theory, we have everything that we need in order to outline this flag. However, some might find it impractical to fabricate this using the inscribed radii as the baseline for these shapes. I have run into that problem, and therefore I have taken the extra step of defining the grown of each layer overall. I would like to know the visible and invisible circumscribed radii of each star, from which I would like to confirm a factor of growth with respect to the central star.

• r_1.n=r_2.nsin126/sin18
  

At this point, I'm stopping, because this is where I have to double-check my math. So far, I have at the very least determined r_2.n. I think that what I have here is the next step for solving r_1.n, for my fabrication needs. My math muscles need a little more basic stretching before I can feel ergonomically confident figuring out the rest. Ah, the days when I used to work on calculus for fun.

One last thing. I have quite easily determined that the triangle ought to be tilted at an angle of sin^-1[17(389^1/2)/389].