Angle of a cone's development sector

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A right cone has base radius $9$ cm and height $12$ cm. Unrolling the lateral surface onto a plane gives a circular sector: what is its central angle in degrees?

Solution

The slant height (sector radius) is $a=\sqrt{9^2+12^2}=\sqrt{225}=15$ cm. In the development the sector arc is the base circle $2\pi r$. The angle is $\alpha=360^\circ\cdot\frac{r}{a}=360^\circ\cdot\frac{9}{15}=216^\circ.$ $\boxed{\alpha=216^\circ}$