A circle has radius 9. What is the length of an arc with a central angle of 80°, in terms of π? (Enter your answer as a coefficient of π;…

An arc is a fraction of the full circumference, and the fraction is the central angle over $360°$. The circumference is $2\pi r = 2\pi(9) = 18\pi$. The fraction is $\frac{80}{360} = \frac{2}{9}$. So the arc length is $\frac{2}{9} \cdot 18\pi = 4\pi$, and the coefficient of $\pi$ is 4. A common wrong answer is 18, the full circumference, with no fraction applied. Another is 36, which comes from using the area formula $\pi r^2 = 81\pi$ times a fraction; the area of a sector is $18\pi$ in that case, which is a different quantity. Arc length uses the circumference, not the area.