dv said:

dv said:

KJW said:

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

To generalise

The base b of the rectangle would be 6 cosθ
The height of the triangle would be 6 sinθ
Pythag:
r^2 = h^2 + (b-r)^2
r^2 = 36 (sinθ)^2 + (6cosθ – r)^2
r^2 = 36 (sinθ)^2 + 36 (cosθ)^2 + – 12cosθr + r^2
36 (sinθ^2 + cosθ^2) = 12cosθr
36 = 12cosθr
r = 3/cosθ

A = r b = 3/cosθ * 6 cosθ = 18

Well I picked up a scale rule, checked the 6 for length/ratio, then measured the rectangle and divided by the ratio to get 17.9.