# How do I calculate the height of an arc?

I'm a hobbyist engineer, having one of those moments where my mind goes blank.
I know this is a simple problem, but I can't remember how to approach it.

I have an arc defined by width and angle. ($w$ and $a$)

The radius ($r$) of the arc is defined by: $$r=\frac{w}{sin(a)}$$

The question is, how do I calculate the height ($h$) of the arc, for any given width ($w$) and angle ($a$)?
The arc is displayed in white, in the diagram below.

In the diagram, the value of $h$ is provided by the computer software, and I need to know how to calculate this value manually.
Additionally, $a$ is specified as $45$ degrees, but this angle may change in the future.

## 2 Answers

From the figure, $w=r\sin a$, \begin{align}h&=r-r\cos a=r(1-\cos a)\\ &=\frac w{\sin a}(1-\cos a)=\frac{w(1-\cos a)(1+\cos a)}{\sin a(1+\cos a)}\\ &=\frac{w\sin^2a}{\sin a(1+\cos a)}=\frac{w\sin a}{1+\cos a}\\ &=w\tan\left(\frac a2\right)\end{align}

Try this:

$$h=r-w\cdot\sin (90-a)$$

Notice $h$ is the difference between the radius and the height of the $79$-$79$-$r$ triangle. That height will be $w\sin\theta$, where $\theta$ is the central angle of that triangle. However, the angle listed as $a$ is complementary to $\theta$, hence $w\sin (90-a)$.