mklotz
Well-Known Member
Recently I was involved in a discussion on another forum concerning how to determine the diameter of a bolt circle given the (chordal) distances between any three points that lie on said bolt circle.
Long ago I wrote a program to do just this calculation. (It can be downloaded from my site - look for CIRC3). However, this revisit to the problem led to a much simpler method that can be easily carried out on any scientific calculator. I thought it might be worthwhile passing it along to others who might encounter this problem.
Call the three chords you've measured a,b,c. They form a triangle and we'll label the angles opposite these sides as A, B, C respectively. Now, the law of cosines says:
a^2 = b^2 + c^2 - 2bc*cosA
from which:
A = acos[(b^2 + c^2 - a^2)/(2bc)]
Since a, b and c are known (measured) quantities, we can plug in their values and find the angle A.
Those who didn't sleep through trig class will also remember the law of sines which says:
a/sinA = b/sinB = c/sinC
If you really, really paid attention in trig class you'll remember this common ratio is also the diameter of the circle which circumscribes the triangle. Thus, since the circumscribed circle corresponds to our bolt circle, the diameter of the bolt circle, D, is given very simply by:
D = a/sinA
Long ago I wrote a program to do just this calculation. (It can be downloaded from my site - look for CIRC3). However, this revisit to the problem led to a much simpler method that can be easily carried out on any scientific calculator. I thought it might be worthwhile passing it along to others who might encounter this problem.
Call the three chords you've measured a,b,c. They form a triangle and we'll label the angles opposite these sides as A, B, C respectively. Now, the law of cosines says:
a^2 = b^2 + c^2 - 2bc*cosA
from which:
A = acos[(b^2 + c^2 - a^2)/(2bc)]
Since a, b and c are known (measured) quantities, we can plug in their values and find the angle A.
Those who didn't sleep through trig class will also remember the law of sines which says:
a/sinA = b/sinB = c/sinC
If you really, really paid attention in trig class you'll remember this common ratio is also the diameter of the circle which circumscribes the triangle. Thus, since the circumscribed circle corresponds to our bolt circle, the diameter of the bolt circle, D, is given very simply by:
D = a/sinA
