The base of a unit equilateral triangle is oriented horizontally. The center of a semicircle is at the midpoint of this base, and the semicircle is tangent to the other two sides of the triangle. Atop the semicircle sits a rectangle whose horizontally oriented base is tangent to the semicircle and whose upper two corners are on the other two sides of the triangle.

What is the maximum area of the rectangle?
Bonus: For extra fun, first guess whether the maximal rectangle will be flatter than a square or taller than a square. Did you guess correctly?