Square Root of 331

√331 at a glance

Exact value

√331

Decimal (10 places)

18.1934053987

Rounded

18.2 · 18.19 · 18.193

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.193405

Prime factorization

331

Cube root

6.917396

How to simplify √331

331 is a prime number, so its only factors are 1 and 331. There is no perfect-square factor to pull out, which means √331 is already in its simplest radical form.

The square root of any prime is irrational. If √331 were a fraction a/b in lowest terms, then a² = 331b², so 331 would divide a — and then 331 would divide b too, contradicting “lowest terms.” That is why the decimal 18.1934053987 is only a rounded value.

Where √331 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √331 lies between 18 and 19. 331 is 7 above 324 and 30 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.1892 (0.02% low)
• Tangent from 18, i.e. 18 + 7 ÷ 36: 18.1944 (0.01% high)
• Tangent from 19, i.e. 19 − 30 ÷ 38: 18.2105 (0.09% high)

For √331 the tangent at 18 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 331 is just 7 above 324.

Finding √331 with the Babylonian method

If a guess is too big, 331 ÷ 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√331) in one step.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √331 = 18.1934053987 to every decimal shown.

√331 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √331 the pattern is [18; 5, 5, 1, 6, 2, 3, 1, 1, 2, 1, 2, 1, …] with the block of 34 terms after the semicolon repeating forever (only the first 12 of the 34 are shown). A pattern that never ends is one more proof that √331 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 − 331y² = 1. Its smallest solution in positive whole numbers is x = 2,785,589,801,443,970, y = 153,109,862,634,573 — 16 digits for x, even though 331 is small, which is what makes Pell’s equation famous.

√331 in geometry and everyday measurements

• A square patio or deck of 331 square feet is about 18.19 ft (18 ft 2 in) on each side, so edging all the way around takes 4 × √331 ≈ 72.8 ft.
• 331 is not a sum of two whole-number squares — 331 is itself a prime that is one less than a multiple of 4, which rules that out — so √331 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 9 × 15 box, because 5² + 9² + 15² = 331.

Square roots near √331 and related roots

• The cube root of 331 is about 6.917396.
• Squaring undoes the root: (√331)² = 331, while 331² = 109,561 — the number whose square root is 331.

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