# Heronian mean

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

${\displaystyle H={\frac {1}{3}}\left(A+{\sqrt {AB}}+B\right).}$

It is named after Hero of Alexandria.

## Properties

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

## Application in solid geometry

A square frustum, with volume equal to the height times the Heronian mean of the square areas

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:

${\displaystyle H={\frac {2}{3}}\cdot {\frac {A+B}{2}}+{\frac {1}{3}}\cdot {\sqrt {AB}}.}$

## 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