This page is an extension of my original tank volume page, a page that analyzes a storage tank with elliptical end caps, lying horizontally. Since I published the original article, I have received a number of inquiries about tanks in other orientations, most often a vertically mounted tank. This page analyzes that tank orientation.

Examine Figure 1, a diagram of a storage tank with elliptical end caps standing vertically. Notice that the diagram consists of a central cylindrical section and two elliptical end caps. The diagram contains most of the variable names and descriptions that will be used in this derivation:

L	The length of the cylindrical section.
R	The radius of the cylinder and the major radius of the elliptical end caps.
r	The minor radius of the elliptical end caps.
y	The height of the tank's contents.

While reading what follows, remind yourself that this is a flat diagram of a three-dimensional object. In particular note that R, the radius of the cylinder and the major radius of the ellipses, describes a circular cross-section extending into a third dimension not shown in this diagram.

This article describes a procedure to compute a vertical tank content volume if given y, the content height within the tank. Unlike the case of a horizontal tank, the vertical form requires an algorithmic solution with three distinct computations, one for each zone.

Elliptical End Caps (applied to zones 1 and 3)

Let's start with the half-ellipses that lie at the bottom and top of the tank. First, we need to get the area of a circle in the horizontal plane that slices through an ellipse at a vertical position given by y. For a spheroid with major radius R and minor radius r, such a function is:

(1)

Now let's integrate this circle area to get a partial ellipse volume:

(2)

Equation (2) completes the volume of an ellipse with major radius R and minor radius r, between the bottom of the tank and any chosen y value within the range 0 <= y <= 2r. With range adjustments, this derivation will be used during volume computation of the bottom and top sections of the tank (zones 1 and 3 in Figure 1).

Center Cylindrical Section (applied to zone 2)

The volume of a cylinder is trivially computed using:

(3)

For our multi-part, zone-oriented solution, in the above equation, H = y-r (see Figure 1).

It must be emphasized that, although the equations for this solution are not particularly complex, the solution is algorithmic, not purely algebraic. Unlike the horizontal tank problem, there is no single equation that can be applied to the entire tank, instead a three-part solution is required, and the selected zone depends on the measured height of the tank's contents.

Here is a table that describes the zones and their corresponding equations:

Zone (see Figure 1)	Range of y values	Solution (volume in units of y cubed)
1 (bottom ellipse)	0 <= y < r
2 (cylindrical section)	r <= y < r+L
3 (top ellipse)	r+L <= y <= 2r+L

It is important to emphasize that each of the zone solutions listed above provides an independent tank volume for the corresponding value of y, e.g. one does not add individual zone volumes to arrive at a solution.

Sanity Check

As readers apply this zone solution, to avoid errors they should apply common-sense tests to their methods. Zones 1 and 3 should each have volumes of 2/3 π R² r, and Zone 2 should have a volume of π R² L. The volume of the entire tank should equal 4/3 π R² r + π R² L.

1. Choose input and output units:

Input Units (length)	Output Units (volume)
Centimeters	Centimeters3
Meters	Meters3
Inches	Inches3
Feet	Feet3
	Liters
	Gallons

2. Enter measured values for the tank:

Variable Name	Description	Value	Units
L	Length of cylindrical section
R	Radius of cylinder / Major radius of ellipse
r	Minor radius of ellipse
y	Volume height argument

3. Compute Volume:  

Variable Name	Description	Value	Units
v	Partial volume of tank for argument y

4. (optional) Create Data Table

Step Size:   

Data Table Area:

(press "Create Table" or "Create Database" above)

Click a link to access these additional resources (or use the drop-down lists at the top and bottom of this page):

• Horizontal Tank Method
• Tank Measurement Issues
• Programming Resources (Java, spreadsheet)
• Reader Inquiries
• Disclaimer