Imagine the circle filled with infinitely many concentric rings.
Now focus on the first quadrant. Imagine straightening all the quarter circles into vertical strips. Why do all the tops of the strips lie on the line $y = \frac{1}{2}\pi x$? Remember there are infinitely many strips so they fill a triangular area as shown in the diagram.
Why is it that the arcs are not stretched in this process, only straightened, so that the area of each strip stays the same when you straighten each arc into a straight line?
What does this tell you about the area of the circle?
The 'strings' could be longer. Suppose instead of straightening quarter circles you straightened arcs which were whole circles. Then on what line would the tops of the straight 'stand up arcs' lie? Again the area of the triangle packed with strips made of straightened arcs of circles is equal to the area of the circle and this gives the formula for the area of the circle.