Calculate the radius of a circle given the chord length and height of a segment.

Question

Please, could some one explain how the first equation becomes the second equation.

1. $a^2 + (r-h)^2=(r\sin{\theta})^2+(r\cos\theta)^2=r^2$

2. The solution to this is expressed as: $r=\frac{a^2+h^2}{2h}$

Many thanks, Martin

Answer 1

\begin{gather*} a^2 + (r-h)^2 = (r \ sin\theta)^2 + (r \ cos\theta)^2 \\ \iff a^2 + r^2 - 2rh + h^2 = r^2(sin^2\theta + cos^2\theta) \end{gather*} From here, we can use the pythagorean identity: $sin^2\theta + cos^2\theta = 1$ (which is trivial to prove if you draw a unit circle) in order to rewrite the right hand side of the equation as just $r^2(1) = r^2$. Hence, we get:

\begin{gather*} a^2 + r^2 - 2rh + h^2 = r^2 \\ \iff a^2 - 2rh + h^2 = 0 \\ \iff 2rh = a^2 + h^2 \\ \iff r = \frac{a^2 + h^2}{2h} \end{gather*}

$Q.E.D.$

Answer 2

Faster: Use the Geometric Mean Theorem („$h^2=pq$“) on the right triangle with half the segment as height and the circle diameter containing the segment height as hypothenusis. That is, $$(a/2)^2=h\cdot(2r-h)$$ And solve for $r$.