Abstract: The alternating genus of a knot is the minimum genus of a surface onto which the knot has an alternating diagram satisfying certain conditions. Very little is currently known about this knot invariant. We study spanning surfaces for knots, and define an alternating distance from the extremal spanning surfaces. This gives a lower bound on the alternating genus and can be calculated exactly for torus knots. We prove that the alternating genus can be arbitrarily large, find the first examples of knots where the alternating genus is exactly 3, and classify all toroidally alternating torus knots.