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Pythagorean Means Calculator (AM, GM, HM)

Calculate the arithmetic, geometric, and harmonic mean of two positive numbers — the three classical Pythagorean means.

Arithmetic mean (AM)
6.5
Geometric mean (GM)
6
Harmonic mean (HM)
5.538462

How it works

These are the three 'Pythagorean means' of two positive numbers a and b, each averaging them in a different sense. Arithmetic mean: AM = (a + b) ÷ 2 — the familiar average. Geometric mean: GM = √(a × b) — the side length of a square with the same area as an a-by-b rectangle. Harmonic mean: HM = 2ab ÷ (a + b) — the arithmetic mean of the reciprocals, reciprocated back.

For any two distinct positive numbers, HM ≤ GM ≤ AM always holds (the AM-GM-HM inequality), with all three equal only when a = b. This calculator computes all three directly from the closed-form formulas above, not an iterative approximation.

FAQ

When should I use the geometric mean instead of the arithmetic mean?

The geometric mean is the right average for quantities that combine multiplicatively rather than additively — growth rates, ratios, or percentage changes across several periods. Averaging a +50% year and a -50% year arithmetically gives 0%, which is misleading (the two together actually leave you down 25%); the geometric mean reflects that correctly. See the CAGR Calculator for exactly this kind of compounding-rate use case.

What's a practical use for the harmonic mean?

The harmonic mean is the right average for rates measured per unit of something else, most classically average speed over equal distances (not equal time) at two different speeds. It's always the smallest of the three means, which reflects that it weights smaller values more heavily than a plain average would.

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