[Math] Calculate the radius of a circle given the chord length and height of a segment

Question

I have a (circular) segment of known height and known chord length. Is is possible to determine the radius of the circle?

Any help much appreciated.

Answer 1

We can apply the Intersecting Chords Theorem.

You chord length is the length $UV$ and the segment height is the length $PX$.

The intersecting chords theorem tells us that $XP \times XQ = XU \times XV$.

Let $\ell = UV$ and $h=XP$. It follows that $UX = XV = \tfrac{1}{2}\ell$. The ICT then tells us that

$$\tfrac{1}{2}\ell \times \tfrac{1}{2}\ell = h \times XQ \, ,$$ i.e. $XQ = \tfrac{1}{4h}\ell^2$. The diameter $PQ=PX+XQ$ and $$PX + XQ = h + \frac{\ell^2}{4h}=\frac{4h^2+\ell^2}{4h}$$

The radius is then one half of this, i.e.

$$CQ = \frac{4h^2+\ell^2}{8h} \, . $$