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Differentiating the arc-length identity gives $f^{\prime }{f}^{\prime \prime }+h^{\prime }{h}^{\prime \prime }=0$. Multiply by $f^{\prime }$ and use $(f^{\prime }{)}^2+(h^{\prime }{)}^2=1$ to obtainThis derivation does not divide by $h^{\prime }$, so it remains valid where $h^{\prime }=0$. Inserting the first fundamental form and second fundamental form computed above gives the Gaussian curvature of a surface of revolution:

The surface is obtained by revolving the graph $z=f(x)$ about the $x$-axis. It has a central cylindrical section of radius $b$, two smooth transition collars, and unit spherical caps centred at $(\pm 3,0,0)$; its projection onto the $(x,z)$-plane is the region $|z|\le f(x)$.

For a surface of revolution, the Gaussian curvature of a surface of revolution isThus the cylindrical region has $K=0$. On each transition collar, $K<0$ where $f^{\prime \prime }>0$, vanishes at the unique inflection circle, and is positive where $f^{\prime \prime }<0$. The spherical caps have $K>0$.