The same bit was where the challenge was for me too, so I did it backwards, starting with the equation and then wondering how to measure the relevant data in the picture.
Something that I also considered was this.
Draw a straight line through the center of the circles, O. It cuts through the inner circle at A and the outer at B.
Using a compass, choose OA as radius and B as center point. It cuts through like OA somewhere past point B. Call that point C.
Generate perpendicular lines s and t with radius OA at points O and C.
Using a compass again, with radius equal to the distance from A to B, mark points D and E on lines s and t, on the same side of line OA.
Draw the rectangle OCED.
The single measurement is the area of the rectangle OCED.
To get the area between the two circles, just multiply the area of the rectangle with pi.
But.. getting the area of a rectangle with a single measurement could be a bit tricky. 🙂
One more variant of the problem would be to find a way to generate a circle with the same area as the space between the two concentric circles, using a straight line and a compass only. With the first solution thought through, it is rather obvious, but I kinda like the classic straight-line-and-compass setup.