An oblique cylinder (also called an inclined cylinder) is tilted: the generators are no longer perpendicular to the bases.
Two different lengths become important:
-
The perpendicular height \(h\): distance between the base planes measured perpendicularly. This controls the volume.
-
The slant height (generator length) \(h_{\text{slant}}\): the length of a side segment connecting corresponding points on the two bases.
This influences the lateral area.
The key fact is that the volume does not change if the base radius \(r\) and the perpendicular height \(h\) stay the same. This is
an instance of Cavalieri’s principle: solids with equal heights and equal cross-sectional areas at every level have the same volume. For a circular cylinder,
every slice parallel to the base is a circle of area \(\pi r^2\), so
\[
\begin{aligned}
V = \pi r^2 h
\end{aligned}
\]
remains valid for both right and oblique cylinders (as long as \(h\) is the perpendicular height).
The lateral area changes because the side surface becomes “stretched.” For an oblique circular cylinder with parallel generators,
the lateral area is still the base perimeter times the generator length:
\[
\begin{aligned}
A_L = (2\pi r)\,h_{\text{slant}}.
\end{aligned}
\]
If you describe the tilt using a horizontal offset \(d\) between the centers of the two bases, then a typical generator forms a right triangle
with legs \(h\) and \(d\). That gives the relationship
\[
\begin{aligned}
h_{\text{slant}} = \sqrt{h^2 + d^2}.
\end{aligned}
\]
The total surface area is then
\[
\begin{aligned}
A_T = A_L + 2\pi r^2.
\end{aligned}
\]
Practical interpretation: volume is about “how much material fits” and depends on the perpendicular stacking height \(h\). Lateral area is about the
length of the side surface you would paint or wrap, and depends on the slant height \(h_{\text{slant}}\).
