The problem with that line of reasoning is that if we continue the line OB so it intersects with the circle at a point F:

BF != BE

I.e. The red area is a quarter of an ellipse, not a circle.

I think the diagram and the law of small numbers might contribute to this faulty line of reasoning (which I followed too!) since BE and BF are jolly close -- if you calculate the area of the red quadrant using your (and my) method, it differs from Bella L's by only 0.02.

If you were to construct the diagram for yourself on a piece of paper and then set your compass at point B, and set the radius to BE, you would then describe a circle that intersected with the larger circle. If you set the compass at point B, and set the radius to BF you would describe a circle that lies inside the larger circle but which doesn't entirely cover the red sector.