Calculate circle radius with segment height and perimeter

[Oppogo]

(mentor note: posted in a non-homework forum hence no template)

Hello!
I have a problem I'm trying to solve.
I'm transforming a circle with known radius. Knowing it's radius i can calculate the circumference.
I transform it by squeezing one side, leveling it, creating a circle segment with a measurable height and the same perimeter as prior circle circumference.
Is there any way to calculate the segments circles radius?
Imagine you can't measure the chorde.
All I'm given is height and perimeter.

 

[Buzz Bloom]

Hi oppogo:

I think this is a neat problem.

I suggest you draw a new diagram extending a line segment from the arc through what you show as the as the height to a point that represent the unknown center of a new circle. Let R be the distance from this center to the arc, that is R is the unkown radius of the new circular arc. Also draw the two radii to the ends of the arc. Let θ be the angle of half the arc. Let A be the length of the arc. Let S be the secant across the arc. r is the old radius.

From this diagram you can write several equations. (There is more than one way to do this.)
1. cos θ = expression₁(R, h)
2. A = expression₂(R, θ)
3. A + S = expression₃(r)
4. S = expression₄(R, h)

ADDED
From these 4 equations you can calculate R. You can first get two equations involving the two unknowns θ and R. One equation is a quadratic and the other a trigonometric. These two equations can be combined to get a single messy equation involving the unknown R and the knowns h and r. This probably will need a numerical solution by a method like successive approximations.

ADDED AGAIN
It may be possible to calculate a good first approximation by assuming h << r.

Hope this helps.

Regards,
Buzz

 

[Oppogo]

Hi oppogo:

I think this is a neat problem.

I suggest you draw a new diagram extending a line segment from the arc through what you show as the as the height to a point that represent the unknown center of a new circle. Let R be the distance from this center to the arc, that is R is the unkown radius of the new circular arc. Also draw the two radii to the ends of the arc. Let θ be the angle of half the arc. Let A be the length of the arc. Let S be the secant across the arc. r is the old radius.

From this diagram you can write several equations. (There is more than one way to do this.)
1. cos θ = expression₁(R, h)
2. A = expression₂(R, θ)
3. A + S = expression₃(r)
4. S = expression₄(R, h)

ADDED
From these 4 equations you can calculate R. You can first get two equations involving the two unknowns θ and R. One equation is a quadratic and the other a trigonometric. These two equations can be combined to get a single messy equation involving the unknown R and the knowns h and r. This probably will need a numerical solution by a method like successive approximations.

ADDED AGAIN
It may be possible to calculate a good first approximation by assuming h << r.

Hope this helps.

Regards,
Buzz

Is there any formula or way to derive radius of the circle segment from only those 2 parameters though? Mathematically, without the need to extend the circle to the imaginary center. That is the problem I'm facing. I have to predict where the center may be located and for now I'm only given these 2 parameters.

 

[Oppogo]

For clarity sake, this is what I'm trying to accomplish. I may be overlooking something really simple but I can't seem to understand how to predict O1 which would be squeezed circles new radius (by extending the arch of circle segment). Thing is, I only know h, r of the circle. What am I missing?

 

[Buzz Bloom]

Hi Oppogo:

I see how you got your first equation ending in: = r π .
I don't see how you got from that to the second equation beginning: sin φ₀ = .
I don't see how you got from that to the third equation beginning: a = .

ADDED
However, I do see how you get the second equation from the diagram.
But I still don't see how you get the third equation from using the first two equations.

Regards,
Buzz

 

[Oppogo]

Hi Oppogo:

I see how you got your first equation ending in: = r π .
I don't see how you got from that to the second equation beginning: sin φ₀ = .
I don't see how you got from that to the third equation beginning: a = .

Regards,
Buzz

These are not my equations. They are from a paper about modeling a peristaltic pump.
Regardless, I already made approximate models of circle segments for descending h values and constant perimeter.
I will look more into this problem tomorrow, for now this will be sufficient.

 

[Buzz Bloom]

Hi Oppogo:

Now I see it. The first equation is not needed.
The second equation is revived from the figure.
The third equation is derived from the second equation. The intermediate step I missed before is:

1- sin φ₀ = h/(r+a).​

Sorry for my confusion.

You still can't calculate a since you don't know φ₀.
However, the first equation can be written

r+a = r π / ( cos φ₀ - (π/2) × φ₀ ),​

and third equation can be written
r+a = h / (1 - sin φ₀ ).
By equating both RHSs you get an equation, and simplifying that you get an equation of the form:

expression (φ₀) = h,​

and this expression has no unknowns other than φ₀.

However I don't think this equation can be solved except by numerical methods. This is the same conclusion I reached in my post #2.

Regards,
Buzz

 

[Buzz Bloom]

Hi @Oppogo:

I do find this a nice problem, and I have continued to think about it. If you are still interested I will post my solution, including the numerical part which involves using a spread sheet like Excel or Livre Office Calc. I think my new solution is easier to understand than the ones discussed here earlier.

Regards,
Buzz