How is $ 2 \pi r=(2 \pi-\theta) R $?

Question

The problem is this : We remove a sector with angle $\theta$ from a circular disk of radius $r$ and center O, and with the remaining part we construct a cone.

Then how did they get that :

$$ 2 \pi r=(2 \pi-\theta) R? $$

Answer 1

Note that in your figure $OA=R$ and not $r$. $r$ is the radius of the circle at the base of the cone. Then the circumference of that is $2\pi r$. But this is equal to the circumference of the original disk, minus the $AB$ sector. So $$2\pi r=2\pi R-R\theta=R(2\pi-\theta)$$