How do you find the radius of an arc using only the rise and arc length?

Answer 1

1. Chord length and rise:

Three points determine a circle, so if you measure the distance

between two points on the circle, then go perpendicular to that line

from the center of the line and measure the distance to the third

point on the circle, you can calculate the radius. If the first

distance is 2*a (divide it by 2 to get a), and the second distance is

b, the rise of the arc

2*a

Then the radius is r = (a^2 + b^2)/(2b). - Pythagoras Theorem

I calculated this by putting the intersection of the chord and line at

(0,0) and plotting your three points as,

(-a, 0), (a, 0), and (0, b).

And then plugging those into the general form of the equation for a

circle, namely

(x - h)^2 + (y - k)^2 = r^2.

I got h = 0, k = (b^2 - a^2)/(2b), and r = (a^2 + b^2)/(2b).

You could use this technique. If you don't have the length

a (or 2a) but you have the length of the curved side (arc length, say c)

(This is what you asked - Radius from Arc length and rise)

then you have a much harder equation to solve for r:

2br = (sin [c/(2r)])^2 + b^2.

But be warned that these computations assume that your curve is a

portion of a circle. If it is some other curve, then there is no radius.