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.

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

Answer 2

I would suggest that you read a book on telescope mirror making. The telescope maker's approximate formula for the radius of curvature of a mirror surface is s = r^2/2R, where r is half the mirror diameter, s is the radius of curvature of the surface (half the focal length), and R is the saggita (depth of the curve). If the mirror is a paraboloid instead of a sphere, this formula is exact, but it is darn close for spherical mirrors.

This problem, with the same solution, comes up in amateur radio: how far can two UHF stations communicate at at sea where the earth is smooth, given the radius of the earth and the height of their antennas.

Or how far the mast of a ship must be to disappear from the horizon, given the height of the mast and the curvature of the earth. Don't ask why Columbus did not do such a measurement.