Square Root of 339

√339 at a glance

Exact value

√339

Decimal (10 places)

18.4119526395

Rounded

18.4 · 18.41 · 18.412

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.411953

Prime factorization

3 × 113

Cube root

6.972683

How to simplify √339

The prime factorization of 339 is 3 × 113. Every prime appears only once, so there is no pair to bring outside the radical — √339 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 339, 3 and 113 appear an odd number of times, so √339 is irrational and 18.4119526395 is a rounded value.

Where √339 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √339 lies between 18 and 19. 339 is 15 above 324 and 22 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.4054 (0.04% low)
• Tangent from 18, i.e. 18 + 15 ÷ 36: 18.4167 (0.03% high)
• Tangent from 19, i.e. 19 − 22 ÷ 38: 18.4211 (0.05% high)

For √339 the tangent at 18 wins, missing by only 0.0047. Tangent estimates shine when the number sits close to a perfect square — here 339 is just 15 above 324.

Finding √339 with the Babylonian method

If a guess is too big, 339 ÷ 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√339) in one step.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √339 = 18.4119526395 to every decimal shown.

√339 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √339 the pattern is [18; 2, 2, 2, 1, 17, 1, 2, 2, 2, 36] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √339 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 − 339y² = 1. Its smallest solution in positive whole numbers is x = 97,970, y = 5,321.

√339 in geometry and everyday measurements

• A square patio or deck of 339 square feet is about 18.41 ft (18 ft 5 in) on each side, so edging all the way around takes 4 × √339 ≈ 73.6 ft.
• 339 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 √339 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 17 box, because 1² + 7² + 17² = 339.

Square roots near √339 and related roots

• The cube root of 339 is about 6.972683.
• Squaring undoes the root: (√339)² = 339, while 339² = 114,921 — the number whose square root is 339.

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