Finding the height above the bottom of a horizontal cylinder (such as a cylindrical gas tank) to which the it must be filled for it to be one quarter full amounts to
plugging 
(one quarter of the area of a full circle) into the equation for the area of a circular segment of radius 
,
 |
This gives equality between the two shaded areas in the above figure, resulting in
 |
where 
and 
is the radius of the circle or cylinder. This cannot be solved analytically, but
the solution can be found numerically to be approximately 
(OEIS A133742),
corresponding to 
.
