# Heronian mean

From HandWiki

In mathematics, the **Heronian mean** *H* of two non-negative real numbers *A* and *B* is given by the formula:

- [math]\displaystyle{ H = \frac{1}{3} \left(A + \sqrt{A B} +B \right). }[/math]

It is named after Hero of Alexandria.

## Properties

- Just like all means, the Heronian mean is symmetric and idempotent.

## Application in solid geometry

The Heronian mean may be 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.

## Relation to other means

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

- [math]\displaystyle{ H = \frac{2}{3}\cdot\frac{A+B}{2} + \frac{1}{3}\cdot\sqrt{A B}. }[/math]

## References

- Bullen, P.S. (2003),
*Handbook of Means and Their Inequalities*, Mathematics and Its Applications (2nd ed.), Berlin, New York: Springer Science+Business Media, ISBN 978-1-4020-1522-9 - Eves, Howard Whitley (1980),
*Great Moments in Mathematics (Before 1650)*, Mathematical Association of America, ISBN 978-0-88385-310-8, https://archive.org/details/greatmomentsinma0007eves

## External links

- Mean-Trapezoids Geometric comparison of some mathematical means

Original source: https://en.wikipedia.org/wiki/Heronian mean.
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