Square Root of 314

√314 at a glance

Exact value

√314

Decimal (10 places)

17.7200451467

Rounded

17.7 · 17.72 · 17.720

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.720045

Prime factorization

2 × 157

Cube root

6.796884

How to simplify √314

The prime factorization of 314 is 2 × 157. Every prime appears only once, so there is no pair to bring outside the radical — √314 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 314, 2 and 157 appear an odd number of times, so √314 is irrational and 17.7200451467 is a rounded value.

Where √314 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √314 lies between 17 and 18. 314 is 25 above 289 and 10 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.7143 (0.03% low)
• Tangent from 17, i.e. 17 + 25 ÷ 34: 17.7353 (0.09% high)
• Tangent from 18, i.e. 18 − 10 ÷ 36: 17.7222 (0.01% high)

For √314 the tangent at 18 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 314 is just 10 below 324.

Finding √314 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 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √314 = 17.7200451467 to every decimal shown.

√314 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √314 the pattern is [17; 1, 2, 1, 1, 2, 1, 34] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √314 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 − 314y² = 1. Its smallest solution in positive whole numbers is x = 392,499, y = 22,150. Because the period is odd, the equation with −1 on the right also has a solution: 443² − 314 × 25² = −1.

√314 in geometry and everyday measurements

• A square patio or deck of 314 square feet is about 17.72 ft (17 ft 9 in) on each side, so edging all the way around takes 4 × √314 ≈ 70.9 ft.
• 314 = 5² + 17², so by the Pythagorean theorem √314 is the diagonal of a 5 × 17 rectangle — and the distance between the points (0, 0) and (5, 17) on a grid.

Square roots near √314 and related roots

• The cube root of 314 is about 6.796884.
• Squaring undoes the root: (√314)² = 314, while 314² = 98,596 — the number whose square root is 314.

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