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# Cylindrical tank volume

##### Cylindrical tank volume
Timur2011-07-30 10:41:00

This is simple calculator which computes volume of liquid in cylindrical tank laying on side. For more advanced version, which computes volume of liquid in tilted tank go to Tilted cylindrical tank volume

##### Cylindrical tank volume

Ok, the solution is simple if you look at the picture. (If there are more than half of cylinder filled with liquid, we can compute air volume and subtract from total.)

Total volume of cylinder is
$V=S_{base}H=\pi R^2H$

So we need to find area of circle filled with liquid and multiply by height.
Filled area is sector area minus triangle area.
$S_x=S_{sect}-S{\Delta}$

Sector area is
$S_{sec}=\frac{\alpha R^2}{2}$, where alpha is arc angle.

Arc angle is unknown yet. To find it, we draw vertical line from center of circle. It divides triangle to two right-angle triangles with hypotenuse equals to R and top cathet R-m.
Thus,
$cos(\frac{\alpha}{2})=\frac{R-m}{R}$
and
$\alpha=2arccos(\frac{R-m}{R})$

The top triange is isosceles. The sides equal to R. Let's find its base. The base is two times cathet from right-angle triangle, which is
$\sqrt{R^2-(R-m)^2}$
according to pythagorean theorem.

Now, knowing all triangle sides we can compute its area using, for example, Calculator of area of a triangle using Hero's formula

$S=\sqrt{p(p-a)(p-b)(p-c)}$
where
$p=\frac{a+b+c}{2}$

Now we know sector area and triangle area. If we subtract triangle area from sector area we get filled area, multiply it by height and get the answer.

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