Square Root of 6

√6 at a glance

Exact value

√6

Decimal (10 places)

2.4494897428

Rounded

2.4 · 2.45 · 2.449

Perfect square?

No — between 2² and 3²

Rational?

Irrational

Both square roots

±2.449490

Prime factorization

2 × 3

Cube root

1.817121

How to simplify √6

The prime factorization of 6 is 2 × 3. Every prime appears only once, so there is no pair to bring outside the radical — √6 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 6, 2 and 3 appear an odd number of times, so √6 is irrational and 2.4494897428 is a rounded value.

Where √6 sits between perfect squares

4 = 2² and 9 = 3² are the nearest perfect squares, so √6 lies between 2 and 3. 6 is 2 above 4 and 3 below 9, so the root is closer to 2.

• Straight line between 4 and 9: 2.4000 (2.02% low)
• Tangent from 2, i.e. 2 + 2 ÷ 4: 2.5000 (2.06% high)
• Tangent from 3, i.e. 3 − 3 ÷ 6: 2.5000 (2.06% high)

For √6 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √6 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 2 (2² = 4):

The count of correct decimals went 1, 3, 7 and all 10 over 4 steps — roughly doubling each time — until the guess matched √6 = 2.4494897428 to every decimal shown.

√6 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √6 the pattern is [2; 2, 4] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √6 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 6y² = 1. Its smallest solution in positive whole numbers is x = 5, y = 2.

√6 in geometry and everyday measurements

• A square tile with an area of 6 square inches has sides about 2.449 in long.
• 6 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √6 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 2 box, because 1² + 1² + 2² = 6.

√6 = √2 × √3, so it shows up whenever those two roots are multiplied — for example the space diagonal of a 1 × 1 × 2 box is √6.

Square roots near √6 and related roots

• The cube root of 6 is about 1.817121.
• Multiplying the radicand by 100 multiplies the root by 10: √600 = 10 × √6 ≈ 24.494897.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.