Find the radius from a chord

[JohnS]

Originally Posted by alexuslara

how about calculating the area of a segment when the only information available are the length of the cord and the hight of the segment? (in your figure it would be "C" and "M"). Thanks, Gabriel

You can't express it easily in those variables. Introduce two variables
R = 0.5*M + 0.125*C²/M
Theta = 2*arcsin(0.5*C/R) Be sure to be in radian mode
then Area = 0.5*R²*Theta

I suppose you can substitute the equation for R in both the theta and area equations, but it would be a mess. I'd calculate the intermediate variables.

using the posted diagram, (m squared plus 1/2c squared) divided by 2m

Ok i am reading but not getting it, maybe someone can help me out understand something. Our business we install frames for theater screens and some have a curve, our guru at getting the cord for us or radius whichever it is has left and no longer works with us. Now when he used to give u the measurements it would be simple, for example the frame is 50ft wide, he would say the ends need to be 10 feet from the backwall and the center of the frame needs to be 7feet from the back wall giving us that nice curve, so my question how the heck does he know those measurements, i think we had one job where they told us either cord was 2.5 which i have no flippin clue what that meant untill one of their engineers told us how to set it. what numbers would i need to be able to determine how much we bring out the ends from the center.?

[JohnS]

Originally Posted by Unregistered

Ok i am reading but not getting it, maybe someone can help me out understand something. Our business we install frames for theater screens and some have a curve, our guru at getting the cord for us or radius whichever it is has left and no longer works with us. Now when he used to give u the measurements it would be simple, for example the frame is 50ft wide, he would say the ends need to be 10 feet from the backwall and the center of the frame needs to be 7feet from the back wall giving us that nice curve, so my question how the heck does he know those measurements, i think we had one job where they told us either cord was 2.5 which i have no flippin clue what that meant untill one of their engineers told us how to set it. what numbers would i need to be able to determine how much we bring out the ends from the center.?

The distance from the backwall at the center is arbitrary as far as the curve is concerned. The difference in distances from the backwall is the h in all the formulas given (10 ft -7 ft = 3 ft in your example). h has to be calculated.

You need to know the radius the theater owner wants (flat is a radius of infinity, which makes h zero, the ends and center are the same distance from the backwall). Then you need to know either the arc length (the width of the screen material, measured along the curve) or the chord (the width of the screen measured in a straight line, not around the curve).

With either of those and the radius, you can calculate h.

Robert, thanks for all the information you posted in here, it help me a lot.
Walter

If the arc and chord lengths are known, can one compute the radius of the circle?

[JohnS]

Originally Posted by Unregistered

If the arc and chord lengths are known, can one compute the radius of the circle?

See post #22. You have to solve a transcendental equation. Ugly.

This is an old problem I had to solve when I was in college - It took me a cubic equation. I had it in a copybook but my wife cleaning the house throw it away thinking it was old rubbish. I can't remember how I solved it.
It is a semicircle in which there are 3 chords, measuring 3, 2 and 1 units. The 3 chords are one after the other, from one to the other extreme of the diameter.
Find the radius.
Many thanks!

[jasoncampbell123]

I'm not going to solve it, but one approach is take 0.0 as the center of the semicircle. The circle is given by x² + y² = R²

At one end of diameter, swing a unit circle, (x+R)² + y² = 1, the intersection of this represents the end of the 1 unit chord. Eliminate y and get x1 = (1-2R²)/(2*R). Plug this into R² equation to get y1, as a function of R.

At the other end of diameter, swing a 3-unit circle. The intersection is the end of the three unit chord. Similarly, determine x3 and y3 as function of R.

Require (x3 - x1)² + (y3-y1)² = 2². This will be some horrible function of R. Solve for R.

An alternate approach, which can only be solved nummerically is to note the sum of the central angles of the three chords is 180° (or pi radians)
2* ASIN(0.5/R) + 2* ASIN(1/R) +2* ASIN(1.5/R) = 180°
From the geometry of the problem, R has to be more than 1.9 and less than 3.
Tabulate for a few values of R, and interpolate between those that give a total closest to 180°. Iterate. 2.056545 =< R =< 2.056546 after a few trials.

[Moncery]

A circle by all three points can also be found in the construction with a compass and ruler. This also provides the position of the center point, and therefore the RADIUS. In the top of the page, the three orange dots can be used with this method. See a circle through three points to build.