Quandaries and Queries


To Whom it concerns:
Can you help me find a formula: find the length of a circle's chord with a known arc length and radius/diameter lengths. The best I have found so far is: there is NO formula for this. Is this true? We've somehow got to at least accurately approximate it. (example: an arc length is This will be used in a homebuilding project and I could use an answer ASAP if someone there knows this. Thanks in advance so much.

a homebuilder




A relationship involving the length of an arc (a), the radius of the circle (r) and the angle at the center of the circle (theta) is

a = r theta

where theta is measured in radians.

Let x be the length of the chord and bisect the angle at the center of the circle to get a right triangle ABC as in the diagram.

Since triangle ABC is a right triangle

 x/2 = r sin( theta /2)

but a = r theta so

x = 2 r sin( a/2r )