The unrolled surface shown in the above figure is in the form of a sector of a sector with radius L, central angle

θ, and length of arc s. The length of arc s is the circumference of the base of the cone which is denoted by c. Thus s = c.

$c = 2 \pi r$

$s = L \theta_{rad}$

$s = c$

$L \theta_{rad} = 2 \pi r$

$\theta_{rad} = \dfrac{2 \pi r}{L}$

$A_L = A_{sector}$

$A_L = \frac{1}{2} sL$

$A_L = \frac{1}{2}(L \theta_{rad})L$

$A_L = \frac{1}{2} L^2 \theta_{rad}$

$A_L = \frac{1}{2} L^2 \left( \dfrac{2 \pi r}{L} \right)$

$A_L = \pi rL$ (*okay!*)