Square Root of 318

√318 at a glance

Exact value

√318

Decimal (10 places)

17.8325545001

Rounded

17.8 · 17.83 · 17.833

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.832555

Prime factorization

2 × 3 × 53

Cube root

6.825624

How to simplify √318

The prime factorization of 318 is 2 × 3 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √318 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 318, 2, 3 and 53 appear an odd number of times, so √318 is irrational and 17.8325545001 is a rounded value.

Where √318 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √318 lies between 17 and 18. 318 is 29 above 289 and 6 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.8286 (0.02% low)
• Tangent from 17, i.e. 17 + 29 ÷ 34: 17.8529 (0.11% high)
• Tangent from 18, i.e. 18 − 6 ÷ 36: 17.8333 (0% high)

For √318 the tangent at 18 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 318 is just 6 below 324.

Finding √318 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √318 = 17.8325545001 to every decimal shown.

√318 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √318 the pattern is [17; 1, 4, 1, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √318 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 − 318y² = 1. Its smallest solution in positive whole numbers is x = 107, y = 6.

√318 in geometry and everyday measurements

• A square patio or deck of 318 square feet is about 17.83 ft (17 ft 10 in) on each side, so edging all the way around takes 4 × √318 ≈ 71.3 ft.
• 318 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 √318 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 14 box, because 1² + 11² + 14² = 318.

Square roots near √318 and related roots

• The cube root of 318 is about 6.825624.
• Squaring undoes the root: (√318)² = 318, while 318² = 101,124 — the number whose square root is 318.

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