# What's the formula for radius given width and height?
**URL:** https://boards.straightdope.com/t/whats-the-formula-for-radius-given-width-and-height/255600
**Category:** Factual Questions
**Created:** [July 17, 2004, 5:16pm UTC](https://boards.straightdope.com/t/whats-the-formula-for-radius-given-width-and-height/255600 "2004-07-17T17:16:59Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@Emilio\_Lizardo](https://boards.straightdope.com/u/Emilio_Lizardo)
#### Post date: [July 17, 2004, 5:16pm UTC](https://boards.straightdope.com/t/whats-the-formula-for-radius-given-width-and-height/255600/1 "2004-07-17T17:16:59Z")
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
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### Author:  [@erislover](https://boards.straightdope.com/u/erislover)
#### Post date: [July 17, 2004, 5:57pm UTC](https://boards.straightdope.com/t/whats-the-formula-for-radius-given-width-and-height/255600/2 "2004-07-17T17:57:56Z")
[Here’s a page](http://www.woodweb.com/knowledge_base/Calculating_RadiansChords.html) that should help you.
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### Author:  [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)
#### Post date: [July 17, 2004, 7:25pm UTC](https://boards.straightdope.com/t/whats-the-formula-for-radius-given-width-and-height/255600/3 "2004-07-17T19:25:44Z")
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.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [July 17, 2004, 9:03pm UTC](https://boards.straightdope.com/t/whats-the-formula-for-radius-given-width-and-height/255600/4 "2004-07-17T21:03:07Z")
> [@Q.E.D.](#):
>
> 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.