## r[{13sin18+n389^(1/2)sin126-nsin18}/(13sin18)]

**Jean Valjean**, in Nerd May 22 2018 · 81 views

- 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 <r
_{2}>, - To coordinates of the star's center are <3r, 10r>.

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 r

_{2}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 r
_{2}. - The angle between them is half of 72
^{o}. - The angle opposite of r
_{2}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 r
_{2}via the Law of Sines. - r
_{2}=r(sin18/sin126)

- 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>.
- h
^{2}=[(-17r)^{2}+(10r)^{2}]^{1/2} - h=r389
^{1/2}

- h
_{2}=r389^{1/2}-r(sin18/sin126) - h
_{2}=r(389^{1/2}-sin18/sin126) - h
_{2}=r(389^{1/2}sin126-sin18)/sin126

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

- 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}=r_{2}+nt - r
_{2.n}=r(sin18/sin126)+nr(389^{1/2}sin126-sin18)/(13sin126) - r
_{2.n}=r[(sin18/sin126)+n(389^{1/2}sin126-sin18)/(13sin126)] - r
_{2.n}=r(13sin18+n389^{1/2}sin126-nsin18)/(13sin126)

- r
_{1.n}=r_{2.n}sin126/sin18

_{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].

24601