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Units in Cylinder, Cone, and Sphere Volume Calculation: 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 of Volume Calculation
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 dropdown menu to select which calculation to perform, you must select whether the
cylinder is more or less than half full.
Variables used in Volume Equations
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 [L^{3}].
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.
References
Rade, Lennart and Bertil Westergren. 1990. Beta Mathematics Handbook. CRC Press. 2ed.
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