Square Root of 211

√211 at a glance

Exact value

√211

Decimal (10 places)

14.5258390463

Rounded

14.5 · 14.53 · 14.526

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.525839

Prime factorization

211

Cube root

5.953342

How to simplify √211

211 is a prime number, so its only factors are 1 and 211. There is no perfect-square factor to pull out, which means √211 is already in its simplest radical form.

The square root of any prime is irrational. If √211 were a fraction a/b in lowest terms, then a² = 211b², so 211 would divide a — and then 211 would divide b too, contradicting “lowest terms.” That is why the decimal 14.5258390463 is only a rounded value.

Where √211 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √211 lies between 14 and 15. 211 is 15 above 196 and 14 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.5172 (0.06% low)
• Tangent from 14, i.e. 14 + 15 ÷ 28: 14.5357 (0.07% high)
• Tangent from 15, i.e. 15 − 14 ÷ 30: 14.5333 (0.05% high)

For √211 the tangent at 15 wins, missing by only 0.0075. Tangent estimates shine when the number sits close to a perfect square — here 211 is just 14 below 225.

Finding √211 with the Babylonian method

If a guess is too big, 211 ÷ 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√211) in one step.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √211 = 14.5258390463 to every decimal shown.

√211 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √211 the pattern is [14; 1, 1, 9, 5, 1, 2, 2, 1, 1, 4, 3, 1, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √211 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 − 211y² = 1. Its smallest solution in positive whole numbers is x = 278,354,373,650, y = 19,162,705,353.

√211 in geometry and everyday measurements

• A square patio or deck of 211 square feet is about 14.53 ft (14 ft 6 in) on each side, so edging all the way around takes 4 × √211 ≈ 58.1 ft.
• 211 is not a sum of two whole-number squares — 211 is itself a prime that is one less than a multiple of 4, which rules that out — so √211 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 9 × 11 box, because 3² + 9² + 11² = 211.

Square roots near √211 and related roots

• The cube root of 211 is about 5.953342.
• Four times the radicand doubles the root: √844 = 2 × √211 ≈ 29.051678.

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