Square Root of 209

√209 at a glance

Exact value

√209

Decimal (10 places)

14.4568322948

Rounded

14.5 · 14.46 · 14.457

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.456832

Prime factorization

11 × 19

Cube root

5.934472

How to simplify √209

The prime factorization of 209 is 11 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √209 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 209, 11 and 19 appear an odd number of times, so √209 is irrational and 14.4568322948 is a rounded value.

Where √209 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √209 lies between 14 and 15. 209 is 13 above 196 and 16 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.4483 (0.06% low)
• Tangent from 14, i.e. 14 + 13 ÷ 28: 14.4643 (0.05% high)
• Tangent from 15, i.e. 15 − 16 ÷ 30: 14.4667 (0.07% high)

For √209 the tangent at 14 wins, missing by only 0.0075. Tangent estimates shine when the number sits close to a perfect square — here 209 is just 13 above 196.

Finding √209 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 209: following the tangent line down to zero simplifies to averaging x with 209 ÷ x.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √209 = 14.4568322948 to every decimal shown.

√209 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √209 the pattern is [14; 2, 5, 3, 2, 3, 5, 2, 28] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √209 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 − 209y² = 1. Its smallest solution in positive whole numbers is x = 46,551, y = 3,220.

√209 in geometry and everyday measurements

• A square patio or deck of 209 square feet is about 14.46 ft (14 ft 5 in) on each side, so edging all the way around takes 4 × √209 ≈ 57.8 ft.
• 209 is not a sum of two whole-number squares — the prime factor 11 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √209 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 12 box, because 1² + 8² + 12² = 209.

Square roots near √209 and related roots

• The cube root of 209 is about 5.934472.
• Four times the radicand doubles the root: √836 = 2 × √209 ≈ 28.913665.

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