Circle from two tangents from an external point

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From an external point $P$ two tangents are drawn to a circle; they form an angle of $60^\circ$. The distance from $P$ to the center is $14$ cm. Find the radius and the length of the circle.

Solution

The radius at the point of tangency is perpendicular to the tangent. The line $PO$ bisects the angle between the tangents, so in the right triangle $O$-tangency-$P$ the angle at $P$ is $30^\circ$: $\sin30^\circ=\frac{r}{\overline{OP}}\ \Longrightarrow\ r=14\cdot\frac12=7\ \text{cm}.$ Length of the circle: $C=2\pi r=14\pi\approx43.98\ \text{cm}.$ $\boxed{r=7\ \text{cm};\ \ C=14\pi\approx43.98\ \text{cm}}$