Question

In an arch, if you have the overall width and height what is the formula to find the radius that creates the arch?

Answer 1

Answer on Question#38265 – Math - Geometry

We have:

Let $h$ be the arch height and $w$ be the arch width. We must find the radius $R$ that creates the arch.

Let's construct a picture:

From the picture:

$$\left(\frac {w}{2}\right) ^ {2} = R ^ {2} - d ^ {2}$$ $$d = R - h$$

Then

$$\frac {w ^ {2}}{4} = R ^ {2} - (R - h) ^ {2} = R ^ {2} - R ^ {2} + 2 R h - h ^ {2}$$ $$w ^ {2} = 4 h (2 R - h)$$ $$2 R - h = \frac {w ^ {2}}{4 h}$$ $$2 R = h + \frac {w ^ {2}}{4 h}$$ $$R = \frac {h}{2} + \frac {w ^ {2}}{8 h}$$

Answer:

$$R = \frac {h}{2} + \frac {w ^ {2}}{8 h}$$