# Radius of circumference as function of arc lenght and height

• jonjacson
In summary, the student is trying to solve a problem in geometry where they do not know the radius of the circumference. They are trying to calculate the radius of the circumference using the arc length and the distance between two points. However, they are getting an error when trying to solve the equation. They have read about the equation for the chord and tried to use that equation, but they get an error. They have also read about the equation for the circular segment and tried to use that equation, but they get an error. They are then trying to solve the problem using Mathematica, but they get an error.
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:

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 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:

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!

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jonjacson said:
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.

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!

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

Code:
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.

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

Thanks

## 1. What is the formula for finding the radius of a circle given the arc length and height?

The formula for finding the radius of a circle given the arc length and height is r = (h^2 + (l/2)^2) / 2h, where r is the radius, h is the height, and l is the arc length.

## 2. Can the radius of a circle be determined if only the arc length or height is known?

Yes, the radius of a circle can be determined if only the arc length or height is known. However, you will also need to know at least one other variable such as the angle or diameter to use the appropriate formula.

## 3. How does the radius of a circle change as the arc length or height increases?

The radius of a circle increases as the arc length or height increases. This is because a larger arc length or height results in a larger circumference, which requires a larger radius to maintain the same curvature.

## 4. What is the relationship between the radius and circumference of a circle?

The radius and circumference of a circle are directly proportional. This means that as the radius increases, the circumference also increases, and vice versa. The relationship can be represented by the formula C = 2πr, where C is the circumference and r is the radius.

## 5. Can the radius of a circle be negative?

No, the radius of a circle cannot be negative. The radius is a measure of the distance from the center of the circle to any point on the circumference, and distance cannot be negative. Therefore, the radius is always a positive value.

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