**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 [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*Similar to above.

_{top}will be < 0."**Reference**

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

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