Math question - on topic

[BRuce]

How does one caculate the diameter of a multi-sided ring. like a 5 - 8 sided ring to use as a landscape barrier around a tree.

BRuce

[Robin Lee]

Bruce -

Rules of thumb....

6-sided object: 8-sided object: 12-sided object: 16-sided object: Width = Diameter ÷ 1.7 Width = Diameter ÷ 2.4 Width = Diameter ÷ 3.7 Width = Diameter ÷ 5.0

Where width is width of stave....

So - for 24" diameter, 6 sides - width of stave is 24/1.7 = 14"

Or - if you want a 3' diameter, 8 staves....36/2.4 = 15"....

Cheers,

Rob

[Robin Lee]

6-sided object: Width = Diameter ÷ 1.7 8-sided object: Width = Diameter ÷ 2.4 12-sided object: Width = Diameter ÷ 3.7 16-sided object: Width = Diameter ÷ 5.0

[BRuce]

thanks for all the input. this will be much easier now. well it is off the garage to start the next project.

BRuce

BRuce wrote:

[Leon]

A little late but I have I have a copy of a miter program that is small and it calculates the angles, lengths, and slope of different thickness wood for any given number of sides. I can e-mail a copy to you if you like.

[Robert Bonomi]

A smart-ass answer would be "take the radius of the ring, and multiply by 2".

Seriously, assuming the sides are all the same length, and you know a) the number of sides, b) the length of a side it's fairly straightforward.

"Imagine" a line from the end of each piece, to the center of the ring. You now have a set of matching isosceles triangles, where each side of the ring is the base of one of the triangles.

The 'vertex angle' of those triangles is (obviously) 360/n degrees, where n is the number of sides.

In plane geometry, the sum of the inside angles of a triangle is 180 degrees. And, for an isosceles triangle, the base angles are equal, so the base angle will be (180-vertex_angle)/2.

Thus, the 'base angle' for each segment of an 'n-sided' ring will be: (180-(360/n))/2) or (90-180/n) This doesn't look meaningful, until you realize you can 'imagine' another line from the midpoint of each side to the center. Which just happens to make a right angle with the side.

Now, simple triginometry applies. we have the 'angle', and the length of the 'adjacent side' (which is _half_ the total length of the side of the ring), and want the length of the 'opposite side'. 'opposite'/'adjacent' is the 'tangent' function, so the length of the "opposite side" is:

tangent(90-180/n)*adjacent_side

This 'opposite side' is the radius of the largest circle that fits _inside_ the ring, so you double it to get the diameter.

If you want the circle that connects the 'points' of the ring, use

adjacent_side/cosine(90-180/n)

to determine the radius of _that_ circle, and, again, double the result for diameter.

Short-cut: if you use the total length of the side of the ring, rather than the length from the end to the point where the vertical connects, you get the diameter directly.

Quickie table of tangents and cosines

sides base angle tan cos 4 45 1.000 .7071 5 54 1.376 .5877 6 60 1.732 .5000 8 67.5 2.414 .3826 10 72 3.077 .3090 12 75 3.732 .2588