Take a really, really small line around the circumference, and lay it down horizontally next to the circle.
Then take another line of circumference, and lay it down horizontally on top of the previous circumference line.
Keep going like that, always laying lines down starting on the left, until you put the center dot on top of the lines.
You will have a right triangle.
The height of the triangle is r and the base of the triangle is 2(pi)(r).
Then the area of the circle = the area of the triangle = (1/2)(2*pi*r)(r) = pi*r^2.
Two pictures, one in silver, one in gold. Take your pick.