A horizontal cylindrical segment is the solid cut from a horizontal cylinder
of length 
and radius 
by a single plane oriented parallel
to the cylinder's axis
of symmetry (i.e., a portion of a horizontal cylindrical tank which is partially
filled with fluid).
For a cut made a height 
above the bottom of the horizontal cylinder
(as illustrated above), the volume 
of the cylindrical segment is given by multiplying
the area of a circular segment
of height 
by the length of the tank 
,
![V(L,R,h)=L[R^2cos^(-1)((R-h)/R)-(R-h)sqrt(2Rh-h^2)],](/images/equations/HorizontalCylindricalSegment/NumberedEquation1.svg) |
plotted above. Note that the above equation gives 
, 
, and 
, as expected. Since a circular
segment is the cross section of the horizontal
cylindrical segment, determining the fraction of the
tank that is full is equivalent to determining the fractional area
of a circle covered by the circular
segment.
Finding the height above the bottom of a horizontal cylinder (such as a cylindrical gas tank) to which it must be filled for it to be one quarter full is sometimes known as the quarter-tank problem.
A related problem about measuring the volume of a cylindrical container is Kepler's wine barrel problem, which maximizes the barrel's total volume subject to a fixed diagonal gauging length rather than determining capacity from a liquid height.
