Square Root of 11

√11 at a glance

Exact value

√11

Decimal (10 places)

3.3166247904

Rounded

3.3 · 3.32 · 3.317

Perfect square?

No — between 3² and 4²

Rational?

Irrational

Both square roots

±3.316625

Prime factorization

11

Cube root

2.223980

How to simplify √11

11 is a prime number, so its only factors are 1 and 11. There is no perfect-square factor to pull out, which means √11 is already in its simplest radical form.

The square root of any prime is irrational. If √11 were a fraction a/b in lowest terms, then a² = 11b², so 11 would divide a — and then 11 would divide b too, contradicting “lowest terms.” That is why the decimal 3.3166247904 is only a rounded value.

Where √11 sits between perfect squares

9 = 3² and 16 = 4² are the nearest perfect squares, so √11 lies between 3 and 4. 11 is 2 above 9 and 5 below 16, so the root is closer to 3.

• Straight line between 9 and 16: 3.2857 (0.93% low)
• Tangent from 3, i.e. 3 + 2 ÷ 6: 3.3333 (0.5% high)
• Tangent from 4, i.e. 4 − 5 ÷ 8: 3.3750 (1.76% high)

For √11 the tangent at 3 wins, missing by only 0.0167. Tangent estimates shine when the number sits close to a perfect square — here 11 is just 2 above 9.

Finding √11 with the Babylonian method

If a guess is too big, 11 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√11) in one step.

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

The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √11 = 3.3166247904 to every decimal shown.

√11 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √11 the pattern is [3; 3, 6] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √11 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 − 11y² = 1. Its smallest solution in positive whole numbers is x = 10, y = 3.

√11 in geometry and everyday measurements

• A square tile with an area of 11 square inches has sides about 3.317 in long.
• 11 is not a sum of two whole-number squares — 11 is itself a prime that is one less than a multiple of 4, which rules that out — so √11 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 3 box, because 1² + 1² + 3² = 11.

Square roots near √11 and related roots

• The cube root of 11 is about 2.223980.
• Four times the radicand doubles the root: √44 = 2 × √11 ≈ 6.63325.

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