Partially Full Cylinder, Sphere, and Cone Volume Calculator

Units: barrel=42 US gallons, cm=centimeter, ft=feet, gallon=US gallon, km=kilometer, m=meter

Equations for Sphere, Cylinder, and Cone Volume (Rade and Westergren, 1990)

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 = Sphere or cylinder diameter [L].
D_bot = Cone bottom diameter [L].
D_top = Cone top diameter [L].
h = Cone height [L].
L = Cylinder length [L].
T = Top width of liquid in cylinder [L].
V = Volume of liquid in cylinder, cone, or sphere [L³].
y = Liquid depth in sphere or cylinder [L].
z = Horizontal to vertical side slope of cone.  z<0 if D_top<D_bot.  z=0 if D_top=D_bot.  z>0 if D_top>D_bot.
θ = Angle representing how full the cylinder is [radians or degrees].  An empty cylinder has θ=0^o,  a cylinder with θ=180^o is half full, and a cylinder with θ=360^o is completely full.

Error Messages

"D must be > 0." Cylinder diameter was entered as ≤ 0; it must be positive.
"D's must be ≥ 0." D_top or D_bot 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. D_bot 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. D_top will be < 0." Similar to above.

Reference

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

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