100 evenly-spaced pts on circle w/ r=1; pick 3 pts to form

[mathmathmath93]

If I place a hundred evenly spaced points on the circumference of a circle, radius 1, and then randomly choose three points from the hundred, how do I determine the expected area of the triangle formed by the three points?

I'm thinking about finding the largest possible triangle and the smallest possible triangle and averaging their areas... But that's probably wrong and I doubt I could figure out how to do that anyway.

Any help would be appreciated it (though just an answer wouldn't much help me - I want to know how to do it)... Especially if it explained how to apply whatever concept is behind this problem to another one of a similar nature! Thanks a bunch.

 

[stapel]

If I place a hundred evenly spaced points on the circumference of a circle, radius 1, and then randomly choose three points from the hundred, how do I determine the expected area of the triangle formed by the three points?

Duplicate post.

Note: I'm not sure, but this "MathWorld" article might prove helpful. I don't know that there is much to glean from this "AskNRICH" thread, but the March 2006 "Ponder This" question (solution provided) might usefully relate to the posted question.

 

[mathmathmath93]

Thanks for your response!
The MathWorld page was useful, but I'd already looked at it before, so...
Oh well.
I guessed 1/2.

 

[TchrWill]

If I place a hundred evenly spaced points on the circumference of a circle, radius 1, and then randomly choose three points from the hundred, how do I determine the expected area of the triangle formed by the three points?

If I am interpeting you correctly:

100 equally spaced points are 3.6º apart.

Number them 1 - 100

Pick any 3 points, for example, 1, 36 and 70 labeled a, b and c

Arc ab = (36 - 1)3.6 = 126º
Arc bc = (71 - 36)3.6 = 126º
Arc ca = 360 - 126 - 126 = 108º

Having the central angles for the 3 arcs and the radius of the arcs, determine the area of each sector formed by each arc and its associated chord.

Having the area of the 3 sectors, subtract from the area of the cirle and you have the area of the triangle formed by the three selected points.



R = sector radius
c = chord length
d = distance from center to chord
h = height of segment
s = arc length
µ = sector entral angle, rad.
Ast = segment area
Asr = sector area

Given R and h: µ = 2arccos[(R-h)/R]

Given R and s: µ = s/R

Given R and d: µ = 2arccos[d/R]

Given R and c: µ = 2arsin[c/2R]

Given d and h: R = d + h

Given s and c: c/2s = [sin(µ/2)/µ]

Given s and d: d/s = [cos(µ/2)/µ]

Given c and h: R = [c^2 + 4h^2]/8h

Given c and d: R = sqrt[(4d^2 + c^2)/2]

Given h and s: h/s = [1 - cos(µ/2)]/µ

Given h and µ: R = h/cos(µ/2)

Given µ and d: R = d/cos(µ/2)

Given c and µ: R = c/2sin(µ/2)

s = Rµ

c = 2Rsin(µ/2)

d = Rcos(µ/2)

h = R[1 - cos(µ/2)]

Ast = R^2[µ - sin(µ)]/2

Asr = µR^2/2