Determine the central angle theta Suppose the ends of the cylindrical storage tank in the figure arc circles of radius $3$ ft and the cylinder is $20$ ft long. Determine the volume of the oil in the tank to the nearest cubic  foot if the rod shows a depth of $2$ ft.
 A: Draw a vertical cross-section of the tank. This is a circle. Let $C$ be the centre of the circle. Draw a horizontal straight line to indicate the level of the oil. Note that this line is $1$ foot below the centre, since the oil has maximum depth $2$.
Let the horizontal straight line meet the circle at $A$ and $B$. 
We want to find $\angle ACB$.  This is probably the angle $\theta$ referred to in the title, but for safety we keep calling it $\angle ACB$. To find this angle, draw a perpendicular from $C$ to $AB$, meeting $AB$ at $M$. 
Since $CM=1$, the angle $ACM$ has cosine equal to $\dfrac{1}{3}$. Now, using a calculator, you can find $\angle ACM$. Double to get $\angle ACB$.  
Now you have enough information to find the area of circular sector that starts at $C$, goes to $A$, then around the circle to $B$, and back to $C$.
From this area, we must subtract the area of $\triangle CAB$ to find the area of cross-section of the oil.
By the Pythagorean Theorem, we have $AM=\sqrt{8}$. So $AB=2\sqrt{8}$, and now you can find the area of $\triangle CAB$: it has base $2\sqrt{8}$ and height $1$.
If you prefer to give your trigonometry a workout, you can alternately find the area of $\triangle CAB$ by using the fact that the area is $\frac{1}{2}(3)(3)\sin\theta$, where $\theta=\angle ACB$. 
Once you have the cross-sectional area of the oil, multiply by the length to find the volume. 
