A cone came from a sector?

A cone with a diameter of $16$ and a height of $15$ would, by Pythagorean's Theorem, have a slant height of $\sqrt{15^2 + (\frac{16}{2})^2} = 17$ . This slant height is also the radius of the original circle.

A cone with a diameter of $16$ would have a base circumference of $16 \pi$ . This circumference is also the arc length of the sector that is cut out of the original circle.

Since the original circle has a radius of $17$ , its circumference is $2 \cdot \pi \cdot 17 = 34 \pi$ . If $x$ is the missing angle, we have a proportion $\frac{x}{360°} = \frac{16 \pi}{34 \pi}$ , which solves to $x = 169 \frac{7}{17}° \approx \boxed{169°}$ .