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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
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
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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