# Tank Math: Full or Empty?

*In the “Know-How” section of the March issue, Steven J. Henkind wrote about how fuel gauges operate and how you can prevent fuel-gauge errors. Here’s the mathematical formula he discussed in the story.*

You can also calculate the amount of fuel in a tank mathematically. For a rectangular tank, the calculation is easy: the overall volume of the tank = Length x Width x Height; if the measured height of the fuel is M% of the height of the tank, then the fuel contained = M% x Overall Volume.

The calculation is more complicated for more complex shapes. The tank shown here is an isosceles-triangular right prism, where L is the length, W is the width, H is the height, and the two (equal) sides are both S. If h is the height of the fuel, the ratio h/H is what a fuel gauge that measures height, not volume, will display. To determine the actual volume for this example, use the following formula:

**Volume = (h/H) ^{2} x ((L x W x H)/2)**.

For example, if L = 4 feet, W = 3 feet, and H = 3 feet, the formula holds that when the tank is full (h=H), Volume = (1)^{2} x (4x3x3)/2 = 18 cubic feet. But, when the fuel gauge reads 1/2, the actual volume in the tank is (1/2)^{2} x (4x3x3)/2 = (1/4) x 18 = 4.5 cubic feet. In other words, although the fuel gauge reads 1/2 full (because of the (h/H) ratio), the tank in the illustration is actually only full. To convert from cubic feet to gallons, use the conversion factor 1 cubic foot = 7.48 U.S. gallons. Note also that this formula applies only to this particular shape, and different formulas are required for other shapes.

*Posted: February 21, 2008*

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