Let h = 2R, where R = radius of semicircle.
Let C = centroid of entire region.

The area of the rectangle is 4R.
Its centroid is 1 unit to the left of C.

The area of the semicircle is πR².
Its centroid is 4R/3π to the right of C.

Since the semicircle and rectangle "balance":
(4R)*(1) = πR²*4R/3π

Solving for R, we get: R = sqrt(6)

Therefore, h = 2*sqrt(6)

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I had expected an answer involving π...
wrong again. (Why am I always surprised?)

An earlier Surprise was the length of one
arch of a cycloid: 8R, where R = radius of
generating circle. We are rolling a circle
-- I expected π² at the very least.
But noooo... it turned out to be the
perimeter of the circumscribed square.

Solution submitted by Soroban