Formula for arc segment

Let's say I take a flat circle (a planar dimension) and cut off the top of the circle in a straight line from point A of the arc to point B of the arc.

The distance from one edge of the circle (A) to the other (B) is 506 feet, that being measured in a straight line from the start of the arc to the end of the arc. Call it the "base" of the arc.

Now, let's assign the height from midline of the straight line to midline of the arc, and set that figure to 27 feet. This is a perpendicular height from the base to the highest point of the arc segment.

How would I calculate the distance from A to B along the curved segment? What is the formula?

I got out my Pocket Ref, but couldn't figure it out.


Reply to
Steve B
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Assuming the height of the arc is "a" and the base is "b", you have: a = 27' b = 506'

The radius of the circle that this arc was cut from is: r = (4a^2+b^2)/(8a) = 1198.85'

If the arc distance is "d", then (in radians): d = arcsin(b/(2r)) * r = 254.9'

If you want degrees: d = (pi/180) * arcsin(b/(2r)) * r

Reply to
Damn Dan

I didn't check my numbers before replying above. I should have seen my silly mistake. The arc length must be longer than the arc base. I dropped a 2 accidentally from the final equation.

The arc length distance *should* be (working in radians): d = 2 * r * arcsin(b/(2r)) = 509.8'

Reply to
Damn Dan

This would be a lot easier to convey with a sketch, but I'll try with text.

call your 27 ft vertical y. Let x = 506/2, half of the chord length. extend the vertical to the center of the arc segment or circle it is a segment of. run another radius from the center to the end of the chord.

You now have a right triangle with sides r-27, x and hypotenuse r. r is the radius of the arc. Using pythagorean theorem and algebra you can find r to be 1198.852.

x/r = sin(theta) where theta is the angle between r and the vertical, so theta = 12.183 deg. The included angle of the arc segment is twice that (by symmetry), or 24.366 deg. The length of the arc segment is then pi*r * 24.366/180 or 509.833.

You could probably crunch this all into a single formula, but it hardly seems worth all the algebra for one shot.

I also did this graphically in AutoCAD (MUCH easier!) and it checks.

Reply to
Don Foreman

This site is useful:

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Reply to
Randy Zimmerman

A straight line that joins the ends of a circular arc is called a chord. If you take any 2 chords, AB and CD say, that cross at E (they don't have to be at right angles to each other), then AE x EB = CE x ED. In your case, AE = 506/2 = 253 = EB, CE = 27 and ED = 2r - 27 where r is the radius of the circle. Therefore 253x253 = 27x(2r-27), and, solving for r gives r = (253x253+27x27)/2x27 = 1198.85 Because the chords are at right-angles, the angle subtended by half the arc AB to the centre of the circle can be found by simple trig; its the angle whose Sine is EB/r = 253/1198.85, which is 12.183 degrees, so the angle subtended by the whole arc AB is twice this, or 12.183x2 = 24.366. The ratio of this to 360 (degrees in whole circle) gives the ratio of the arc length to the circumference of the circle. So the arc length is

2xPIx1198.85 x 24.366/360 = 509.83
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Just curious--are you designing a sheet-metal roof, or what?

Reply to
Leo Lichtman

this should help you out .. .. ..

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Reply to

I have been getting a lot of help on this, and I appreciate it.

What I need to figure is street distance where the street is long and curved. The street is part of an imaginary circle, and the two parallel lines that make the street will be of slightly different lengths.

I want to measure in a straight line, the segment of the circumference of the outer longest arc, then measure the height, then apply a formula that will tell me the distance of the curved line.

I use aerial maps, therefore, I cannot always get center points, or some of the other measurements, as they are none, or I could be only guessing, and not exact as I need to be.

Some have suggested using a percentage of the circumference, but I'm limited in what I can measure. But I can always measure the outer arc length and the height.

Hope this helps. I know math pretty well, but because of a brain injury a few years ago, have to now relearn things. I'm starting to get a grasp on this, and know once I get it, it will stay there if only for a few weeks.

I know trig and algebra. I need to read up some and understand radians, and other terms used to try to give me the answer.

Thanks again, and I'm getting there.


Reply to
Steve B

The (free) program, CSEG, on my page will calculate all the variables associated with this sort of problem. All you have to do is input any two pieces of information you know, e.g., chord length and sagitta.

Regards, Marv

Home Shop Freeware - Tools for People Who Build Things

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Reply to

The answer that lemel_man posted yesterday looks correct to me. That should give you all the math you need. Did you not see it? Here it is:

lemel_man wrote:

Condensing all that down to a two formulas and simplifying a bit gives us:

If the straight line distance you measure is C (cord length), and the height is H, then the formula is:

Diameter: D = C*C/4H + H

Arc Length: L = 2*PI*D*arcsin(C/D)/360

For your numbers:

C = 506 H = 27

D = 506*506/4*27 + 27 = 2397.70 L = 2*3.1415*2397.70*arcsin(506/2397.70)/360 L = 15064.7 * arcsin(0.21104)/360 L = 15064.7 * 12.183 / 360 L = 509.82

Reply to
Curt Welch

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