# How to find the radius of a circle that intersecs two adjacent corners and touches the opposite side of a rectangle?

Several times over the years I have needed to use a radius pivot string to help scribe an arc onto a board. This may be a doorway or a ceiling arch. In my latest project I needed to construct a 177" concrete screed with a two inch arc. Using Google Sketchup through trial and error I have come up with a radius of ~69.5 feet. Unfortunately, in Sketchup the circle is generated from the center so it takes several attempts to stumble upon the correct distance. A formula would be very convenient. I've read up on circle trig but getting nowhere.

• Is the opposite side touched parallel to the line formed by adjacent corners? – qwr Oct 9 '16 at 7:10

Is this the situation? The rectangle has length $a$ and height $b$ and the circle with radius $r$ passes through $2$ adjacent corners and touches the other side.

If so, the radius can be found as follows:

$$(r-b)^2 + (\frac {a}{2})^2 = r^2$$ $$=>\;\; r = \frac{a^2 + 4b^2}{8b}$$

If arch radius is known, then to compute arch height $h$ and co-ordinates of arch you can use :

$$h ( 2 R - h) = a^2 ; \quad y = \sqrt{x ( 2 R-x) }$$ 