Circle Area Derivation

When a straight line segment is moved along its perpendicular direction, it sweeps an area.  The rate the area increases with respect to distance moved is merely the length of the line segment.

By direct analogy, the rate that a circle's area increases wrt radius is equal to its circumference.  (Each point on the circle moves perpendicular to its tangent line as radius increases.)

In the following diagram, if the vertical side of the right triangle is constructed equal to the circumference, then the height of each point on the hypotenuse is equal to the circumference of a circle with a radius equal to its projection on the horizontal side.

Imagine constructing the circle and the triangle simultaneously from the center of the circle out.  At each radius, the circle and the triangle are increasing in area at exactly the same rate.  The final area of the triangle is, therefore, equal to the final area of the circle.