Every chemical dose, every flush, every glycol charge and every expansion-tank calculation starts from one question: how much water is in there? Two geometries answer almost all of it.

A run of pipe is a cylinder: $V = \dfrac{\pi D^{2}}{4} L$ — V equals pi D squared over four, times L. $V$ is the internal volume, $D$ the inside diameter (never the nominal size, never the outside), and $L$ the developed length including every riser and drop. The $/4$ is not optional — $\pi D^{2}$ is four times the area of a circle, and forgetting it is the single most common way a dosing calculation ends up four times over strength. Work in metres and the answer is cubic metres; multiply by 1000 for litres.

A horizontal cylindrical tank, part full, is the awkward one: $V = L\left[r^{2}\cos^{-1}\!\left(\dfrac{r-h}{r}\right) - (r-h)\sqrt{2rh-h^{2}}\right]$. Here $L$ is the shell length, $r$ is the radius (half the diameter — the formula wants r everywhere, and mixing in a diameter breaks it silently), and $h$ is the liquid depth measured from the bottom of the shell, running from 0 to $2r$. The bracket is the area of a circular segment; multiplying by the length drags that area down the tank to make a volume.

Here is the nugget the whole lesson exists for. Volume does not follow depth. A horizontal barrel is narrow at the bottom, widest at the centreline, narrow again at the top — so a stick wet to a quarter of the diameter means about 19.5% of capacity, and a stick wet to three quarters means about 80.4%. Below halfway the volume runs behind the depth; above halfway it runs ahead; and only at exactly half do the two agree. Read a horizontal tank as a straight percentage and you will over-order at the bottom and over-fill at the top, every single time.