Quandaries and Queries
  


Name: Dale

Who is asking: Parent
Level: Secondary

Question:
How do I find the properties of a circle that is drawn around an irregular polygon of (n) sides with the lenghts of each side given and all end points of the polygon lye on the circumferance of the circle?



Hi Dale,

For the inscribed triangle whose sides are a, b, and c, the nicest way to write the formula for the circumradius R is to use the semiperimiter s = (a+b+c)/2. Then

R = abc/sqrt[s(s-a)(s-b)(s-c)].

Although there might be a similar formula for the circumradius of an inscribed quadrangle, I do not recall having seen it. My guess is that it would not be a practical formula. Here is how to compute R using a computer:

Notation: AD = a, AB = b, BC = c, CD = d. Use the fact that opposite angles sum to 180 degrees, and that cos A = - cos(180 - A). Then the law of cosines applied to the two triangles that share the diagonal BD gives cos A = (a2 + b2 -c2 - d2)/(2(ab + cd)).

From this we compute BD = sqrt(a2 + b2 - 2ab cos(A))

Finally, use the triangle formula on the triangle whose sides are a, b, and BD.

I see no reason to even think about the formula for the circumradius of an inscribed 5-sided polygon.

Chris


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