# Math Help - Volume of Fuel versus Height of Fuel column in a Cylinderical Horizontal Fuel tank

1. ## Volume of Fuel versus Height of Fuel column in a Cylinderical Horizontal Fuel tank

Hi
Consider a Cylindrical Fuel tank of a Truck. The Cylinder is mounted horizontally which means the axis of the cylinder is horizontal to the ground. Let us say the truck is filled with Fuel to the Full Fuel tank level and the truck starts its long journey. The truck moves consuming Fuel as it travels. Let us assume that the truck travels till such time the Fuel in the Fuel tank is emptied. The cross section of the Fuel column is a Full circle when the Fuel is full and it is half circle when the Fuel is at half fuel tank capacity. The cross section of Fuel column is a segment of the circle at any point in time.

Let us say "h" is the height of Fuel column from the horizontal Bottom. The height of fuel column is equal to Diameter of the cylinder "D" when the fuel column is Full and is Zero when fuel is emptied. Let us have "L" as length of the cylinder.
I want to know for a given "h" what is the "volume of the Fuel" in the Fuel tank. I also want to use the formula in excel sheet and draw a graph between "h" and "volume of Fuel". can someone help me?

2. Let r be the radius of the circle. You need to compute the area of a segment of the circle, which can be determined by calculating (if $h \leq r$) $\theta = \arccos \left(\frac{r-h}{r}\right)$. Then the area of the segment is $\theta r^2 - (\sin \theta)(r-h)r$, where $\theta$ is in radians. If $h \geq r$, then $\theta = \arccos \left(\frac{h-r}{r}\right)$, and the area of the segment is $(\pi - \theta)r^2 + (\sin \theta)(r-h)r$.

3. Thanks icemanfan