Square Root of 309

√309 at a glance

Exact value

√309

Decimal (10 places)

17.5783958312

Rounded

17.6 · 17.58 · 17.578

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.578396

Prime factorization

3 × 103

Cube root

6.760614

How to simplify √309

The prime factorization of 309 is 3 × 103. Every prime appears only once, so there is no pair to bring outside the radical — √309 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 309, 3 and 103 appear an odd number of times, so √309 is irrational and 17.5783958312 is a rounded value.

Where √309 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √309 lies between 17 and 18. 309 is 20 above 289 and 15 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.5714 (0.04% low)
• Tangent from 17, i.e. 17 + 20 ÷ 34: 17.5882 (0.06% high)
• Tangent from 18, i.e. 18 − 15 ÷ 36: 17.5833 (0.03% high)

For √309 the tangent at 18 wins, missing by only 0.0049. Tangent estimates shine when the number sits close to a perfect square — here 309 is just 15 below 324.

Finding √309 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 309: following the tangent line down to zero simplifies to averaging x with 309 ÷ x.

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 √309 = 17.5783958312 to every decimal shown.

√309 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √309 the pattern is [17; 1, 1, 2, 1, 2, 4, 1, 1, 1, 8, 6, 1, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √309 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 − 309y² = 1. Its smallest solution in positive whole numbers is x = 64,202,725,495, y = 3,652,365,444.

√309 in geometry and everyday measurements

• A square patio or deck of 309 square feet is about 17.58 ft (17 ft 7 in) on each side, so edging all the way around takes 4 × √309 ≈ 70.3 ft.
• 309 is not a sum of two whole-number squares — the prime factor 3 and 103 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √309 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 17 box, because 2² + 4² + 17² = 309.

Square roots near √309 and related roots

• The cube root of 309 is about 6.760614.
• Squaring undoes the root: (√309)² = 309, while 309² = 95,481 — the number whose square root is 309.

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