Heronian Mean Calculator

Input two numbers to instantly compute their Heronian Mean (HM).

Calculate the Heronian Mean of Two Numbers

Result

What Is the Heronian Mean, and How Is It Calculated?

The Heronian Mean is a method for averaging two numbers, often used to smooth out the differences between them. The formula for the Heronian Mean is: \( H = \frac{x + y + \sqrt{x \cdot y}}{3} \) Where:

\( x \) and \( y \) are the two numbers to be averaged.
\( \sqrt{x \cdot y} \) is the geometric mean of the two numbers.

Calculation Steps

Sum the Numbers: Calculate \( x + y \).
Find the Geometric Mean: Compute \( \sqrt{x \cdot y} \).
Compute the Heronian Mean: Add the sum and the geometric mean, then divide by 3.

Examples

Example 1: Calculate the Heronian Mean of 8 and 18

Solution:

1. Calculate the sum of the numbers:

\( 8 + 18 = 26 \)

2. Calculate the geometric mean:

\( \sqrt{8 \times 18} = \sqrt{144} = 12 \)

3. Calculate the Heronian Mean:

\( H = \frac{26 + 12}{3} = \frac{38}{3} \approx 12.67 \)

Result: The Heronian Mean of 8 and 18 is approximately 12.67.

Example 2: Calculate the Heronian Mean of 5 and 20

Solution:

1. Calculate the sum of the numbers:

\( 5 + 20 = 25 \)

2. Calculate the geometric mean:

\( \sqrt{5 \times 20} = \sqrt{100} = 10 \)

3. Calculate the Heronian Mean:

\( H = \frac{25 + 10}{3} = \frac{35}{3} \approx 11.67 \)

Result: The Heronian Mean of 5 and 20 is approximately 11.67.