Heronian mean

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The Heronian mean H of two non-negative real numbers A and B is given by the formula:

H = \frac{1}{3}(A + \sqrt{A B} +B)\,.

It is named after Hero of Alexandria, and used in finding the volume of a frustum of a pyramid or cone. The volume is equal to the product of the height of the frustum and the Heronian mean of the areas of the opposing parallel faces.

The Heronian mean of the numbers A and B is a weighted mean of their arithmetic and geometric means:

H = \frac{2}{3}\cdot\frac{A+B}{2} + \frac{1}{3}\cdot\sqrt{A B}\,.

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