KJW said:

KJW said:

KJW said:

18

Actually, this is not correct. In fact, there is insufficient information to obtain a correct numerical answer. My answer was based on the assumption that there is sufficient information to answer the question, and under this assumption, the only possible answer is 18. That’s because the assumption implies that the area is constant with respect to the angle the 6 unit line makes to the diameter. I actually tested that by looking at two particular angles, and for those two angles, the area is indeed 18. But when I examined the area for arbitrary angle, it is not constant. But because the angles I chose were special, they just happen to both lead to an area of 18.

The area is 18(cos²θ + sinθ cosθ)

The angles I chose to test the assumption were θ=45° and θ=0°, both leading to an area of 18, but it should be clear that the area isn’t 18 for other angles.

I get 18 for theta = 30 as well.

The base b of the rectangle would be 6 cos30 = 3 sqrt(3)
The height of the triangle would be 6 sin 30 = 3
Pythag:
r^2 = 3^2 + (b – r)^2
r^2 = 3^2 + (3 sqrt(3) – r)^2
r^2=- 9 + 27 -6 sqrt(3)r + r^2
r = 2 sqrt(3)

A = r b = 18