Horizontal Cylindrical Segment
The solid cut from a horizontal cylinder of length L and radius R 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) is called a horizontal cylindrical segment.
For a cut made a height h above the bottom of the horizontal cylinder (as illustrated above), the volume V(L,R,h) of the cylindrical segment is given by multiplying the area of a circular segment of height h by the length of the tank L,
![画像: V(L,R,h)=L[R^2cos^(-1)((R-h)/R)-(R-h)sqrt(2Rh-h^2)],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HorizontalCylindricalSegment/NumberedEquation1.svg){kind=link}
plotted above. Note that the above equation gives V(h=0)=0, V(h=R)=piR^2L/2, and V(h=2R)=piR^2L, 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 the it must be filled for it to be one quarter full is sometimes known as the quarter-tank problem.
See also
Circular Segment, Cylinder, Cylindrical Segment, Quarter-Tank ProblemExplore with Wolfram|Alpha
More things to try:
Cite this as:
Weisstein, Eric W. "Horizontal Cylindrical Segment." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HorizontalCylindricalSegment.html