Square Root of 319

√319 at a glance

Exact value

√319

Decimal (10 places)

17.8605710995

Rounded

17.9 · 17.86 · 17.861

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.860571

Prime factorization

11 × 29

Cube root

6.832771

How to simplify √319

The prime factorization of 319 is 11 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √319 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 319, 11 and 29 appear an odd number of times, so √319 is irrational and 17.8605710995 is a rounded value.

Where √319 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √319 lies between 17 and 18. 319 is 30 above 289 and 5 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.8571 (0.02% low)
• Tangent from 17, i.e. 17 + 30 ÷ 34: 17.8824 (0.12% high)
• Tangent from 18, i.e. 18 − 5 ÷ 36: 17.8611 (0% high)

For √319 the tangent at 18 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 319 is just 5 below 324.

Finding √319 with the Babylonian method

If a guess is too big, 319 ÷ 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√319) in one step.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √319 = 17.8605710995 to every decimal shown.

√319 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √319 the pattern is [17; 1, 6, 5, 1, 4, 3, 1, 3, 4, 1, 5, 6, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √319 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 − 319y² = 1. Its smallest solution in positive whole numbers is x = 12,901,780, y = 722,361.

√319 in geometry and everyday measurements

• A square patio or deck of 319 square feet is about 17.86 ft (17 ft 10 in) on each side, so edging all the way around takes 4 × √319 ≈ 71.4 ft.
• 319 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √319 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √319 as its space diagonal.

Square roots near √319 and related roots

• The cube root of 319 is about 6.832771.
• Squaring undoes the root: (√319)² = 319, while 319² = 101,761 — the number whose square root is 319.

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