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How do you calculate arc length given length of chord and distance of chord to arc through the center?
It is much easier to follow what is below if you have a rough sketch.
Suppose the length of the chord is 2x units so that half the chord is x units.
Suppose that the distance from the centre of the circle to the arc = y units.
Suppose the angle subtended at the centre by the chord is 2k radians.
Then the semi-chord and the line from the centre form a right angled triangle with the radius to the end of the chord, and the angle subtended by the semi-chord at the centre is k radians.
Now, by Pythagoras the radius, r = sqrt(x2 + y2) units
and tan(k) = x/y (or sin(k) = x/r
Since both x and y are given, r and k can be calculated.
Then arc length = r*k which is a simple multiplication.
If you measure angles in degrees, remember that pi radians = 180 degrees.
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