Heronian mean

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

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

It is named after Hero of Alexandria.

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:

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

External links

Mean-Trapezoids Geometric comparison of some mathematical means

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