Are two angles of a cone the same? If not, what is relationship between them?

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Are the angles $\theta$ and $\alpha$ the same? link

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They are not the same, but related via $r$, the radius of the cone base,

$$\theta L = 2\pi r =2\pi L\sin \frac\alpha2$$

or, $$\theta = 2\pi \sin\frac\alpha2$$