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View Full Version : What's the formula for radius given width and height?

Emilio Lizardo
07-17-2004, 12:16 PM
A few years a go, I saw the following formula in a home repair magazine: given the width and height of a section of arc, it solved for the radius of the circle. This is great for carpenters and designers. I can't find the damned magazine anywhere, though. Do any mathematicaly inclined dopers know this one

erislover
07-17-2004, 12:57 PM

Q.E.D.
07-17-2004, 02:25 PM
The formula in my copy of Engineering Formulas is r = h/2 + s2/8.h, where r is the radius, s is the segment length (linear) and h is the height of the arc.

ultrafilter
07-17-2004, 04:03 PM
The formula in my copy of Engineering Formulas is r = h/2 + s2/8.h, where r is the radius, s is the segment length (linear) and h is the height of the arc.

To be clear, that's r = h/2 +s2/(8h) (which is what Q.E.D. said, but I thought it might be confusing).

Derivation: You got a circle centered at O with radius r, and an arc running from A to B. You also have segment AB, with length s.

Let C be the midpoint of segment AB, and let D be the midpoint of arc AB. Draw OD, which has length R. Let CD have length h, so OC has length r - h. AC and CB both have length s/2. More importantly, OC is perpendicular to AB.

By the law of cosines, in a triangle with side lengths a, b, and c, the equation c2 = a2 + b2 - 2ab*cos(t), where t is the measure of the angle opposite the side of length c.

Here, we'll take t = 90o, a = r - h, b = s/2, and c = r. Recalling that cos(90o) = 0, we plug those values into the formula to get r2 = (r - h)2 + s2/4. Solve for r, and you get the formula above.