Ellipsoid Volume

An ellipsoid is a sphere that has been stretched by different amounts along three perpendicular directions, and its volume is beautifully unfussy: $V = \tfrac{4}{3}\pi abc$, where $a$, $b$ and $c$ are the half-lengths of those three axes. Set all three equal to $r$ and it returns the sphere's $\tfrac{4}{3}\pi r^3$ exactly, as it must. The contrast with the ellipse is instructive: its area is the equally simple $\pi ab$, yet neither the ellipse's perimeter nor the ellipsoid's surface area has any elementary closed form at all.

Earth itself is the most-measured ellipsoid we have, an oblate spheroid about 21 km fatter through the equator than through the poles, and every GPS fix you take is computed against a reference ellipsoid rather than a sphere. On a rather different scale, radiologists estimate the volume of a tumour, a kidney or a bladder from three ultrasound measurements using $V = \tfrac{\pi}{6}LWH$. That is not a different formula, it is this one written with full axis lengths instead of half ones, since $\tfrac{4}{3}\pi\left(\tfrac{L}{2}\right)\left(\tfrac{W}{2}\right)\left(\tfrac{H}{2}\right) = \tfrac{\pi}{6}LWH$.

Which brings up the one reliable trap: semi-axes, not axes. Enter diameters where the formula expects radii and your answer is eight times too large. If you measured an egg as 60 mm long and 45 mm wide, the numbers this solver wants are 30 and 22.5.