Hello, galactus!

A cylindrical barrel of diameter $\displaystyle d$ lies on a stand

so that its axis makes an angle of 20° with the horizontal.

The amount of fluid in the barrel just covers the bottom of the barrel.

Approximate the volume of fluid in the barrel. Code:

D
*
* *
L * * d
* *
* *
* *
A* - - - - - - - - - - - - - - - - - - - - *C
:*::::::::::::::::::::::::::::::::::* :
: *:::::::::::::::::::::::::::* :
: *d:::::::::::::::::::* L :
:20°*:::::::::::::* :
: *::::::* 20° :
P+ - - * - - - - - - - - - - - - - - - - - +Q
B

Let $\displaystyle L$ = length of the cylinder.

In $\displaystyle \Delta APB\!:\;\;AP \:=\:d\cos20^o$

In $\displaystyle \Delta CQB\!:\;\;CQ \:=\:L\sin20^O$

Since $\displaystyle CQ = AP,\;\;L\sin20^o \:=\:d\cos20^o \quad\Rightarrow\quad L \;=\;d\cot20^o$

The fluid occupies __half__ the volume of the cylinder: .$\displaystyle V \;=\;\frac{1}{2}\cdot\pi\left(\frac{d}{2}\right)^2 L$

Therefore: .$\displaystyle V \;=\;\frac{1}{2}\cdot\pi\cdot\frac{d^2}{4}(d\cot20 ^o) \;=\;\frac{\pi}{8}d^3\cot20^o$