Solving for radius and angle given height and width of an arc

Question

Trying to solve for the radius $r$ and angle $\theta$ on the image below given the length dimensions $a$ and $b$.

Attempting to draw a nice curve on a thermal modelling software (IES VE), but I need the radius and angle. Tempted to just use an alternative software to draw the curve and just import it, but due to failed attempts, quite curious myself how this is solved.

Please have a look and looking forward to your responses. Thanks

https://drive.google.com/file/d/1HcfmeOVEA9T5M2eOfCvvETB2BrLh-3oM/view?usp=sharing

Answer 1

Let $a$ and $b$ meet in $C$ and the circle in $A$ and $B$, resp. Now $\theta$ and $r$ won’t change if $C$ was any point of the Thales circle over $AB$. Hence all we know is the length of the cord $AB$ and from here $$\sqrt{a^2+b^2}=2r\sin(\theta/2).$$

Answer 2

The chord length of a circular segment of radius $r$ and central angle $\theta$ is $$c = 2r \sin\frac\theta2 = r\sqrt{2(1-\cos\theta)}$$

In your case we have $c = \sqrt{a^2+b^2}$ and $\theta = \frac{\pi}2 - (\alpha + \beta)$ so

$$\sqrt{a^2+b^2} = r\sqrt{2(1-\sin(\alpha+ \beta))}$$

We get $$r = \sqrt{\frac{a^2+b^2}{2(1-\sin(\alpha+ \beta))}}$$