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

The formula in my copy of *Engineering Formulas* is r = h/2 + s[sup]2[/sup]/8[sup].[/sup]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 +s[sup]2[/sup]/(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 c[sup]2[/sup] = a[sup]2[/sup] + b[sup]2[/sup] - 2ab*cos(t), where t is the measure of the angle opposite the side of length c.

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