Found an answer to my own question. Wording of quote describing Archimedes proof regarding area of a spherical cap is correct, but associated figure confused me. I suggest improving the figure so that the various angles, radii, etc. are clear. As an example, theta is given as a/2, but "a" is not identified anywhere in the figure (or the text as I recall). The radius of the circle is as described in the quote, but it is not shown in the figure, which left me to my interpretation of the figure and the wrong "r", while "h" is depicted in the figure but has no relevance to the Archimedes proof.