Radius of circumference as function of arc lenght and height

[jonjacson]

Homework Statement

I don't know the radius of the circumference. I could only measure the arc lenght, and the height. I know the guidelines say we should not post images, but this is a geometric problem and I think it is something logic to show it with a picture.

So the variables are the distances Radius R of the circumference, distance ce also called sagitta, and arc length of the circumference from point d to point b.

I know the arc length from d to b and the distance ce, and I want to calculate the radius R of the circumference.

Homework Equations

Basic trigonometric functions:

Arc= Radius * Angle

Cos $\alpha$ = adjacent side/ hypotenuse

The Attempt at a Solution

Well, I get an equation, but I don't know to solve it. I hope you can check if my calculations are right and then if there is any method to get the answer R.

Distance ca= Radius R
Distance ce= It is known, it is the sagitta.
Distance ea= R - ce

And now looking at the triangle aeb, I want to calculate the angle between ea and ba sides.

Simply using cosine function:

cos $\alpha$ = (R - ce)/ba = (R - ce)/R

So the angle $\alpha$ = ArcCos ( (R - ce)/R)

Now I will use the equation Arc= Radius * Angle, in this case:

Arc= from point d to point b
Angle= angle between sides da and ba, which is equal to two times the angle $\alpha$ we have just calculated.

So we have:

Arc= Radius * 2*ArcCos((R-ce)/R)

And the problem is that I have R inside the argument of the trigonometric function, and at the same time it is multipliying the trigonometric function, so I cannot get an equation of R as function of ce and the Arc, I don't know how to proceed.

I tried to solve it with Mathematica but I got an error, even if I used a numerical solver. At this point I don't know how to solve R.

Thanks to everybody!

 

[SteamKing]

I know the guidelines say we should not post images, but this is a geometric problem and I think it is something logic to show it with a picture.

Posting images for illustrative purposes is OK under the PF guidelines. If you check other threads, especially in the HW forums, there are many images attached to posts.

The following PF Guideline prohibits posting images or links to obscene material:

Attachments & Links: Images, material or links to images and or material whether real, satirical or implied depicting obscene, indecent, lewd, pornographic, violent, abusive, insulting, or threatening in nature are not permitted on this bulletin board. This includes Gifs or cartoons.

 

[SteamKing]

Check out the geometry of a circular segment:

http://en.wikipedia.org/wiki/Circular_segment

 

[jonjacson]

Thanks SteamKing, I didn't know that.

And talking about the problem I am still struggling to get an equation for R. I see that they introduce the equation for the chord, but since I didn't measure it I cannot use that equation to get the answer.

I apologize for that, maybe this is not my best day, maybe I am not clever enough or it is more difficult than it looks. I don't know.

I have read all the equations at wikipedia and at Wolphram http://mathworld.wolfram.com/CircularSegment.html but I cannot find R as function of the height and the arc length.

Any idea? Any tip? I don't know, Do you think I could solve it using some kind of algebraic manipulation with those equations?

Thanks!

 

[Bill Simpson]

Since you said you tried to use Mathematica, here is one way of solving this using your equations.

In[1]:= ce = 2; arc = 6;
FindRoot[arc - r*2*ArcCos[(r - ce)/r], {r, 3}]

Out[3]= {r -> 1.75044}

(* now check the calculation and see if it seems right *)

In[4]:= r*2*ArcCos[(r - ce)/r] /. {r -> 1.7504436047214182`, cd -> 2}

Out[4]= 6.

This appears to be sensitive to choosing a starting value for r that is close to a solution. Otherwise it ends up finding a complex root and that isn't what you are looking for.

 

[jonjacson]

Well I got an error in Mathematica, but thanks for replying and giving me another way to do it.

Thanks