# Heronian mean

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

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

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:

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