Volume of partially full cylindrical, spherical, and conical tanks Calculations and equations for partially full storage tanks:  Partially full cylindrical tank on its side, partially full spherical container, and conical shape.

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Discussion
This web page is designed to compute volumes of storage tanks for engineers and scientists; however, it may be useful to anyone who needs to know the volume of a full or partially full sphere, cylinder, or cone.  Technically, our "cone" is not really a cone; it is a "frustum of a right cone" (Rade and Westergren, 1990, p. 73).  A cone is a frustum that has a top diameter of 0.0.  For the cylinder laying on its side, if you enter liquid top width, diameter, and length, there are two possible solutions for depth and volume; therefore, when using the drop-down menu to select which calculation to perform, you must select whether the cylinder is more or less than half full.

Variables
D = Cylinder diameter [L].
Dbot = Cone bottom diameter [L].
Dtop = Cone top diameter [L].
h = Cone height [L].
L = Cylinder length [L].
T = Top width of liquid in cylinder [L].
y = Liquid depth in sphere or cylinder [L].
z = Horizontal to vertical side slope of cone.  z<0 if Dtop<Dbot.  z=0 if Dtop=Dbot.  z>0 if Dtop>Dbot.
Ø = Angle representing how full the cylinder is [radians or degrees].  An empty cylinder has Ø=0o,  a cylinder with Ø=180o is half full, and a cylinder with Ø=360o is completely full.

Error Messages
"D must be > 0."  Cylinder diameter was entered as <= 0; it must be positive.
"D's must be >= 0."  Dtop or Dbot for the cone was entered as negative.
"h must be >= 0."  Cone height was entered as negative.
"L must be > 0."  Cylinder length was entered as <= 0.
"T must be >= 0."  Cylinder top width was entered as negative.
"T must be <= D."  Cylinder liquid top width cannot be greater than cylinder diameter.
"y must be >= 0."  Sphere or cylinder liquid depth was entered as negative.
"y must be <= D."  Sphere or cylinder liquid depth cannot be greater than diameter.
"Infeasible. Dbot will be < 0."  The slope that was input for the cone gives a bottom diameter that is negative, which is physically impossible.  Most likely the slope was entered with an incorrect sign.  Slope is Horizontal/Vertical and is negative for a cone with a smaller top than bottom.
"Infeasible. Dtop will be < 0."  Similar to above.

References
Rade, Lennart and Bertil Westergren.  1990.  Beta Mathematics Handbook.   CRC Press.  2ed.