Take the bottom half of the unit circle and cut it into many wedges, like thin slices of pie. Place these wedges next to each other, points up, like the bottom teeth of a wild animal. Split the top half of the circle into wedges pointing down, the top teeth of the animal. Now close the animal's mouth. The teeth interlock perfectly. As the number of teeth increases the shape approaches a rectangle. The width is π (half the circumference), and the height is 1. Therefore the area of the circle is π.
You have to do a little more work to make the above rigorous. For large n, the floor and ceiling of the bounding rectangle approach, but never quite reach, 1 unit apart, and the left and right walls approach π from above. At the same time, the dimensions of the inner rectangle approach 1 and π from below. This is similar to lower and upper sums. The actual area is trapped between two conferging limits, hence the area is π.
The area is also an integral, and if we scale the circle, the area is scaled accordingly. If an ellipse has semimajor and semiminor axes a and b, its area is πab. Set a = b = r and the area of a circle with radius r is πr2.
If a circle is divided into n wedges, each wedge holds 1/n of the area, and cuts 1/n of the circumference. The area of a wedge of the unit circle is always 1/2 of the arc length.