struggling pre-calculus student here, I'm studying for a test and I was wondering how you would go about finding the circumference of a circle when you only have the perimeter and a radian. The prompt I was given was:
"A slice of pizza has a perimeter of $28.2$ inches and the angle formed by the slice is $\dfrac{2π}9$ radians $(40^\circ)$. What is the circumference of the pizza?"
On
Assume that this is the slice of pizza in question.
You know the length of the arc and the angle between the two sides of equal length, i.e., the radius. The perimeter $p$ of the slice is given by $$p = \mathrm{radius} + \mathrm{radius} + \mathrm{length~of~arc}$$ The length of the arc is a product of the angle between the two sides of the slice of the pizza and the radius. Let $r$ denote the radius, and $\theta$ the angle between the two sides. So, $$p = r + r + \theta r\implies r= \frac p{2 + \theta} = \frac{28.2}{2 + \frac{2\pi}9} = \frac{126.9}{12.14} = 10.45$$ The circumference $C$ of the pizza is then $$C = 2\pi r = 2\pi\times 10.45 = 65.67\text{ inches.}$$
Hint: Here, perimeter = $r + r + \theta r$