Mean Proportional Calculator

The mean proportional between two numbers is the value that appears twice in the middle of a proportion. This calculator finds it in exact surd form, and also handles the third and fourth proportional — the two related cases that are easy to mix up.

Formula x = √(a × b)

Try an example
The distinction

Mean, third and fourth proportional

These three names describe where the unknown sits, and confusing them is the usual reason an answer comes out wrong. The pattern is easiest to see written down together.

Mean proportional

The unknown is repeated in both middle positions. You know the two outer terms and want the one that fits between them multiplicatively.

a : x = x : b → x = √(ab)

4 : x = x : 9 gives x = 6, because 4 × 9 = 36 and √36 = 6.

Third proportional

The repeated term is one you already know, and the unknown is the last position. You are continuing a pattern forwards.

a : b = b : c → c = b² / a

4 : 6 = 6 : c gives c = 9, because 6² ÷ 4 = 36 ÷ 4.

Fourth proportional

Nothing repeats. Three terms are known and the fourth completes the proportion — the ordinary case.

a : b = c : d → d = bc / a

4 : 6 = 10 : d gives d = 15, because 6 × 10 ÷ 4 = 60 ÷ 4.

Write the proportion with the unknown in place

For the mean proportional that is a : x = x : b. Writing it out is the step that prevents the mix-up, because you can see immediately whether the unknown appears once or twice.

Cross multiply

a × b = x × x, which is x² = ab. The unknown appearing twice is exactly why a square root turns up — nothing else in proportion theory produces one.

Multiply the two known terms

4 × 9 = 36. If this product is negative there is no real answer, since no real number squares to a negative.

Take the positive square root

x = √36 = 6. If the product is not a perfect square, pull out the largest square factor: √48 = √(16 × 3) = 4√3. That surd is the exact answer.

Check both ratios reduce to the same value

4/6 = 0.667 and 6/9 = 0.667. Equal, so x is right. This also shows why the mean proportional is the multiplicative midpoint: the same factor gets you from a to x as from x to b.

x² = ab  ⟹  x = √(ab)

Mean proportional. The only formula in proportion theory that involves a root, and it comes from the unknown appearing on both sides.

√(ab) ≤ (a + b) / 2

The AM–GM inequality. The geometric mean never exceeds the arithmetic mean, with equality only when a = b. This is why the two markers on the number line above only ever coincide for equal inputs.

The geometric mean theorem

Drop a perpendicular from the right angle of a right triangle to the hypotenuse. It splits the hypotenuse into two pieces, and the height of that perpendicular is exactly the mean proportional between them. Ancient geometers used this to build a square equal in area to any given rectangle.

Worked examples

Perfect squares and surds

A perfect square

Find the mean proportional between 4 and 9.

Proportion
4 : x = x : 9
Cross multiply
x² = 4 × 9 = 36
Square root
x = √36 = 6
Check
4/6 = 0.667, 6/9 = 0.667 ✓
Compare average
(4+9)/2 = 6.5

x = 6 exactly

Surd answer

Find the mean proportional between 3 and 16.

Proportion
3 : x = x : 16
Cross multiply
x² = 3 × 16 = 48
Factor out squares
√48 = √(16 × 3)
Simplify
x = 4√3
Decimal
≈ 6.9282

x = 4√3 ≈ 6.9282

Third proportional

Find the third proportional to 4 and 6.

Proportion
4 : 6 = 6 : c
Cross multiply
4c = 6 × 6 = 36
Divide by 4
c = 36 ÷ 4
Check
4/6 = 0.667, 6/9 = 0.667 ✓

c = 9

Rectangle into a square

What side length gives a square with the same area as a 3 cm × 12 cm rectangle?

Rectangle area
3 × 12 = 36 cm²
Square area
s² = 36
So s is
√36 = 6
Which is
the mean proportional of 3 and 12

s = 6 cm

Reference

Mean proportional values

Pairs whose product is a perfect square give whole-number answers. Everything else gives a surd.

a b Product ab Mean proportional Arithmetic mean
14422.5
281645
3123667.5
493666.5
5201001012.5
663666 — equal
236√6 ≈ 2.44952.5
316484√3 ≈ 6.92829.5
712842√21 ≈ 9.16529.5
1/28424.25
Background

Why a multiplicative midpoint is useful

The ordinary average answers "what number is the same distance from both?" The mean proportional answers "what number is the same factor from both?" Whenever the quantities you are dealing with multiply rather than add, the second question is the one that matters — and asking the first gives a misleading answer.

Growth rates are the clearest case. If an investment gains 100% one year and loses 50% the next, the arithmetic mean of +100% and −50% is +25%, which would suggest a substantial gain. In fact you are exactly where you started: the growth factors were 2 and 0.5, and their mean proportional is √(2 × 0.5) = 1, meaning no change at all. The geometric mean of the factors gives the true average, because returns compound multiplicatively.

Where the classical name comes from

Greek mathematics organised proportions by the position of the repeated term, which is where "mean" and "third" proportional get their names — the mean is the term in the middle, the third is the term that comes third in the sequence. Euclid's construction for the mean proportional is still the standard compass-and-straightedge method for converting a rectangle into a square of equal area, which was one of the fundamental problems of classical geometry.

Reading the surd rather than the decimal

When the answer is 4√3, writing it as 6.9282 loses information. The surd tells you the answer squares back exactly to 48; the decimal only squares to 47.99997. In geometry problems this matters, because a length of 4√3 often combines with other surds to give a clean final answer that would be invisible if you had rounded early. This is why the calculator shows the simplified surd first.

Related tools

If the unknown appears only once in your proportion, you want the cross multiplication calculator instead — no square root is involved. For proportions whose terms are fractions, the proportion calculator with fractions keeps everything exact. And if you are scaling a shape rather than finding a midpoint, the scale factor calculator covers the area and volume relationships that follow from a linear scale.

Questions

Mean proportional — questions

Including the difference from an ordinary average, which is the point most often missed.

What is the mean proportional between two numbers?
The mean proportional between a and b is the number x that makes a : x = x : b — the same value appears in both middle positions. Cross multiplying gives x² = ab, so x = √(ab). Between 4 and 9 the mean proportional is √36 = 6, and indeed 4 : 6 = 6 : 9, since both reduce to 2/3.
Is the mean proportional the same as the geometric mean?
For two numbers, yes — they are the same quantity under two names. "Mean proportional" comes from the theory of proportions and "geometric mean" from statistics, but both mean √(ab). The geometric mean generalises to more than two numbers as the nth root of their product, whereas mean proportional is normally only used for exactly two.
How is the mean proportional different from the average?
The ordinary average (arithmetic mean) is (a + b) / 2 and sits halfway between the two numbers by addition. The mean proportional sits halfway by multiplication — the same factor takes you from a to x as from x to b. For 4 and 9: the average is 6.5, the mean proportional is 6. The mean proportional is never larger than the average, and they are equal only when a = b.
What is the third proportional?
The third proportional to a and b is the number c such that a : b = b : c — here the repeated term is the one you already have, and you want what comes after it. Cross multiplying gives c = b² / a. The third proportional to 4 and 6 is 36 / 4 = 9, which is the same relationship as the first example read in the other direction.
What is the fourth proportional?
The fourth proportional to a, b and c is the d that completes a : b = c : d, giving d = bc / a. Nothing is repeated here, so it is the ordinary proportion problem — the cross multiplication calculator handles the same thing if that framing is easier.
Can the mean proportional be a surd?
Very often. √(ab) is only a whole number when the product ab is a perfect square. Between 3 and 16 the product is 48, and √48 = 4√3 ≈ 6.928. This calculator gives the simplified surd as well as the decimal, because the surd is the exact answer and the decimal is only an approximation.
Can the mean proportional be negative?
Strictly, x² = ab has two roots, +√(ab) and −√(ab), and both satisfy the proportion algebraically. By convention the mean proportional means the positive root, which is the one that makes sense for lengths and other physical quantities. If both a and b are negative their product is positive and a real mean proportional exists; if exactly one is negative the product is negative and there is no real answer.
Where is the mean proportional used in geometry?
In a right triangle, the altitude drawn to the hypotenuse is the mean proportional between the two segments it creates — the geometric mean theorem. It also appears in the construction of a square with the same area as a given rectangle: the side of that square is the mean proportional between the rectangle's length and width. That is the classical reason the quantity mattered, since it solves "squaring the rectangle" with compass and straightedge.