Square Root of 12

√12 at a glance

Exact value

2√3

Decimal (10 places)

3.4641016151

Rounded

3.5 · 3.46 · 3.464

Perfect square?

No — between 3² and 4²

Rational?

Irrational

Both square roots

±3.464102

Prime factorization

2² × 3

Cube root

2.289428

How to simplify √12

Look for the largest perfect square that divides 12. Here it is 4 (2²), because 12 = 4 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 12 = 2² × 3. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 stays inside.

Check: (2√3)² = 2² × 3 = 4 × 3 = 12. As a decimal, 2√3 = 2 × 1.7320508076 ≈ 3.4641016151.

Where √12 sits between perfect squares

9 = 3² and 16 = 4² are the nearest perfect squares, so √12 lies between 3 and 4. 12 is 3 above 9 and 4 below 16, so the root is closer to 3.

• Straight line between 9 and 16: 3.4286 (1.03% low)
• Tangent from 3, i.e. 3 + 3 ÷ 6: 3.5000 (1.04% high)
• Tangent from 4, i.e. 4 − 4 ÷ 8: 3.5000 (1.04% high)

For √12 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 √12 with the Babylonian method

Picture a rectangle with an area of 12 and one side x; the other side must be 12 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √12.

Start from the nearest whole number, 3 (3² = 9):

The count of correct decimals went 1, 3, 8 and all 10 over 4 steps — roughly doubling each time — until the guess matched √12 = 3.4641016151 to every decimal shown.

√12 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √12 the pattern is [3; 2, 6] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √12 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 − 12y² = 1. Its smallest solution in positive whole numbers is x = 7, y = 2.

√12 in geometry and everyday measurements

• A square tile with an area of 12 square inches has sides about 3.464 in long.
• 12 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 √12 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 2 box, because 2² + 2² + 2² = 12.
• Since √12 = 2√3, a length of √12 is exactly 2 copies of the length √3 laid end to end.

Square roots near √12 and related roots

• The cube root of 12 is about 2.289428.
• Four times the radicand doubles the root: √48 = 2 × √12 ≈ 6.928203.

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