Sagitta from arc length and chord length?

[DLJ]

Given you have a particular arc length, a particular chord length, and given that it is a minor circular arc, there is only one circle you can derive from it. I know this is possible, to derive a sagitta (arc height) from the chord length and the arc length, I'm just not sure as to how. Everything I've read thus far states it's not possible, but being a man of science, this is not an acceptable answer. There is a direct correlation between the arc length and chord length to produce the sagitta, there has to be. If you keep a constant chord length of say.. Ten, and you have an arc length of twelve, or fifteen, or five hundred seventy-six, the sagitta will adjust accordingly, so, this tells me there is a direct correlation. That being said, has anyone solved this? Surely I can't be the first person to pose this question and relentlessly hunt for an answer.

In specific, if I have an arc length of 80, and a chord length of 71, on a minor arc, what is the sagitta?
If I can solve for the sagitta with the two variables (a = arc length, c = chord length), I can solve for radius, diameter, circumference, and more importantly, the angle of the arc.

Thank you for reading, and thank you even more if you can help.

-Dj

 

[TchrWill]

Given you have a particular arc length, a particular chord length, and given that it is a minor circular arc, there is only one circle you can derive from it. I know this is possible, to derive a sagitta (arc height) from the chord length and the arc length, I'm just not sure as to how. Everything I've read thus far states it's not possible, but being a man of science, this is not an acceptable answer. There is a direct correlation between the arc length and chord length to produce the sagitta, there has to be. If you keep a constant chord length of say.. Ten, and you have an arc length of twelve, or fifteen, or five hundred seventy-six, the sagitta will adjust accordingly, so, this tells me there is a direct correlation. That being said, has anyone solved this? Surely I can't be the first person to pose this question and relentlessly hunt for an answer.

In specific, if I have an arc length of 80, and a chord length of 71, on a minor arc, what is the sagitta?
If I can solve for the sagitta with the two variables (a = arc length, c = chord length), I can solve for radius, diameter, circumference, and more importantly, the angle of the arc.

Thank you for reading, and thank you even more if you can help.\]\

At least two of the following should lead you to your solution.

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

Reference: Machine Design, August 22, 1985
Robert Dieckann
Grand Island, NE

 

[pka]

Given you have a particular arc length, a particular chord length, and given that it is a minor circular arc, there is only one circle you can derive from it. I know this is possible, to derive a sagitta (arc height) from the chord length and the arc length, I'm just not sure as to how. Everything I've read thus far states it's not possible, but being a man of science, this is not an acceptable answer. There is a direct correlation between the arc length and chord length to produce the sagitta, there has to be. If you keep a constant chord length of say.. Ten, and you have an arc length of twelve, or fifteen, or five hundred seventy-six, the sagitta will adjust accordingly, so, this tells me there is a direct correlation. That being said, has anyone solved this? Surely I can't be the first person to pose this question and relentlessly hunt for an answer

This webpage should answer those questions and more.