Find the lateral area of a cone. Why is it

Pi * r * l

The book I am reading says it is 1/2 the circumference of the base times the slant height. I cannot picture how or why this would get you the lateral area. Can someone help me visualize and understand this?

Looking at the net of the lateral area it looks like 1/3 of a whole circle where the slant height represents that circles radius. So (1/3)(pi * l ^2) would make sense to me.