Here's a neat method I found on woodgears.ca:
Let A and B be the endpoints of your circular arc, and let M be the midpoint between A and B. If P is the highest point on your arc (the apogee), then let h be the distance from M to P. Finally, let L be the distance between A and B:
Note: Considering the triangle formed by the center point, point M, and point A (or B), you have, from the Pythagorean theorem, that r2 = (L/2)2 + (r-h)2. Solving for r, you get r = (L2 + 4h2)/(8h), which could be too large for a typical drawing method (long string with a pencil, or a beam compass).
Rather than working with a super long radius, you can use a fact of geometry: the inscribed angle formed between two points of a circle (A, and, B) and any other point along the perimeter, is constant. Set two nails at points A and B, and get two straight sticks, each a bit longer than L, butted against each other at an angle.
You can use the following image as a guide. Rest the sticks (depicted by the pairs of colorful lines) on the arc-side of the work piece, so each stick touches a nail, and cross the inside edge of the sticks at point P. At that point, the yellow lines are your sticks, the bright green point at their intersection is where you hold the pencil.
Now, holding a pencil at the intersection of the sticks, and keeping the angle the same between the sticks, slide the connected sticks left and right until the pencil has reached both nails. For instance, to draw the right half of the arc, you would move from the yellow position, to the orange position, then the dark orange, etc., until you reach B. You will have drawn a circular arc between A and B with required height h.
This will work as long as the angle between the two sticks is kept constant (see theory here). You would record the angle at point P (because you can precisely calculate that point's position), and then if you keep the same angle when sliding, you will "reveal" your other arc points. You could glue the sticks at the angle you want, or use a protractor to ensure the angle is kept constant.