Geometric Mean Calculator

The geometric mean of two positive numbers a and b is √(a × b) — the square root of their product. It is the "multiplicative average" of two values, in contrast to the arithmetic mean (sum / count). The geometric mean appears in three important places: multiplicative growth problems (compound interest, scale factors), the right triangle altitude theorem (the altitude to the hypotenuse is the geometric mean of the two hypotenuse segments), and as the middle term of a 3-term geometric sequence. This tutorial covers all three.

The definition

For two positive numbers a and b:

Geometric mean = √(a × b)

Note: a and b must both be positive (or both negative — but the result of √(negative × negative) is the same as positive ones). Geometric mean of mixed-sign numbers is undefined for real values.

Arithmetic mean vs geometric mean

For two positive numbers, the geometric mean is always less than or equal to the arithmetic mean (the AM-GM inequality):

√(a × b) ≤ (a + b) / 2, with equality only when a = b.

This is one of the foundational inequalities in mathematics. Equality only when the two numbers are equal (e.g. both = 5: GM = √25 = 5, AM = (5+5)/2 = 5).

Worked example 1 — basic GM

Geometric mean of 4 and 9: GM = √(4 × 9) = √36 = 6.

Compare arithmetic mean: AM = (4 + 9) / 2 = 6.5. GM < AM, as expected.

Notice 4, 6, 9 form a 3-term geometric sequence with common ratio 6/4 = 1.5 and 9/6 = 1.5. The geometric mean of the first and last term is the middle term.

The right triangle altitude theorem

This is where the geometric mean shines in geometry. In a right triangle with the right angle at vertex C, draw the altitude from C to the hypotenuse. This altitude divides the hypotenuse into two segments — call them p (adjacent to one leg) and q (adjacent to the other).

Then three Geometric Mean relationships hold simultaneously:

1. Altitude: h = √(p × q). The altitude is the geometric mean of the two hypotenuse segments.
2. Leg 1 (length a): a = √(p × c), where c = p + q is the full hypotenuse. The leg is the geometric mean of its adjacent segment and the whole hypotenuse.
3. Leg 2 (length b): b = √(q × c). Same as above for the other leg.

These three relationships are the right triangle altitude theorem, sometimes called the "geometric mean theorem" or "Euclid's theorem" (Proposition II.14 of his Elements).

Worked example 2 — altitude theorem

Right triangle ABC with right angle at C. The altitude from C to hypotenuse AB meets AB at point D, dividing AB into segments AD = 4 and DB = 9.

Altitude CD = √(4 × 9) = √36 = 6.

Leg AC = √(4 × 13) = √52 ≈ 7.21. (Here c = 4 + 9 = 13.)

Leg BC = √(9 × 13) = √117 ≈ 10.82.

Verify with Pythagorean: AC² + BC² = 52 + 117 = 169 = 13² = c². ✓

Why does the altitude theorem work?

The altitude from the right angle creates three similar triangles: the original right triangle, and two smaller right triangles formed inside it. All three are similar by AA (each shares the right angle plus another angle from the original).

Corresponding sides of similar triangles are proportional. The altitude theorem expresses these proportionalities in geometric-mean form.

Worked example 3 — geometric sequence middle term

What number, when inserted between 8 and 50, forms a 3-term geometric sequence?

The middle term is the geometric mean: GM = √(8 × 50) = √400 = 20.

Verify: 8, 20, 50 has ratio 20/8 = 2.5 and 50/20 = 2.5. ✓ Geometric sequence with common ratio 2.5.

The geometric mean of n numbers

The two-number case generalizes. For n positive numbers x₁, x₂, ..., xₙ:

GM = (x₁ × x₂ × ... × xₙ)^(1/n) — the nth root of the product.

For 3 numbers: GM = ∛(x₁ × x₂ × x₃). For 4 numbers: GM = ⁴√(x₁ × x₂ × x₃ × x₄). And so on.

The geometric mean has units the same as the values (not units²), unlike the geometric mean from the altitude theorem.

Real-world applications

• Compound annual growth rate (CAGR). When growth rates differ year to year, the "average annual growth rate" is the geometric mean, not the arithmetic mean. A stock that grows 20% one year and 10% the next has an average growth of √(1.2 × 1.1) ≈ 14.89%, not (20 + 10)/2 = 15%.
• Photography. The "average" of two f-stops (which are multiplicative) uses geometric mean. f/2.0 and f/8.0 have a geometric average of √(2 × 8) = f/4.0.
• Aspect ratios. Standard photographic and screen aspect ratios are often geometric means of common ratios (e.g. ISO 216 paper sizes use √2 as the consistent length-to-width ratio).
• Engineering — load testing. Endurance test cycles use geometric means to characterize fatigue ratings.

When to use GM vs AM

Common mistakes

• Using AM where GM is needed. For multiplicative quantities (interest rates, growth factors), arithmetic averaging gives the wrong "average". GM is correct.
• Computing geometric mean of negatives. GM = √(a × b) requires a × b > 0. With mixed signs, the result is imaginary and meaningless in real-world contexts.
• Confusing the altitude theorem variants. Three different geometric means apply (altitude, leg 1, leg 2). Make sure you use the right one for the value you want: h uses both segments; a leg uses one segment and the full hypotenuse.
• Forgetting GM ≤ AM. This is a useful sanity check — if your GM exceeds your AM, you computed something wrong.