Fish Cage Geometry: Finding Formula for a & b

[weedannycool]

We are trying to get a formula for a and b (them being the same lenght). we know length c, circumference of the circle and the length of the arc between a and b, radius of cirle and we think we could work out the angle between the lines.

Thanks in advance

Danny

 

[zgozvrm]

By "formula for a and b," I assume you mean that you're looking for a formula to find the length of a and b, given that you already know the radius of the circle, the length of c and the arc length between a and b.

Also, I assume that c is perpendicular to the circle. That is, if you were to extend c, it would pass through the center of the circle.


Given these assumptions, try this:
1) let 'r' be the length of the radius of the circle
2) let 'x' be the point where 'a' intersects the circle
3) let 'y' be the point where 'c' intersects the circle
4) let 'w be the point at the center of the circle
5) let 'v' be the point of intersection of 'a', 'b', and 'c'
6) Draw a line segment from 'w' to 'x'
7) Draw a line segment from 'w' to 'y' (which is collinear with 'vy')
8) this will result in triangle 'vxw' with sides of the following lengths: a, r, and c+r
9) now, draw a line segment from 'x' to 'y'
10) this will subdivide triangle 'vxw' into to triangles: triangle 'vxy' and triangle 'wxy'
11) since you know the arc length from 'x' to 'y' (half that of the given arc length), you can find angle 'xwy'
12) triangle 'wxy' is isosceles since 'xw' = 'r' and 'yw' = 'r', therefore you can determine angles 'wxy' and 'wyx'
13) you can also determine length of 'xy'
14) and you can determine angle 'xyv'
15) you now have 2 sides ('xy' and 'vy') and the included angle 'xyv', so you can find the third side, 'vx' which is the length of 'a'


I'll leave it to you to do the algebra and come up with the formula.
Good luck...