Square Root of 218

√218 at a glance

Exact value

√218

Decimal (10 places)

14.7648230602

Rounded

14.8 · 14.76 · 14.765

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.764823

Prime factorization

2 × 109

Cube root

6.018462

How to simplify √218

The prime factorization of 218 is 2 × 109. Every prime appears only once, so there is no pair to bring outside the radical — √218 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 218, 2 and 109 appear an odd number of times, so √218 is irrational and 14.7648230602 is a rounded value.

Where √218 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √218 lies between 14 and 15. 218 is 22 above 196 and 7 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.7586 (0.04% low)
• Tangent from 14, i.e. 14 + 22 ÷ 28: 14.7857 (0.14% high)
• Tangent from 15, i.e. 15 − 7 ÷ 30: 14.7667 (0.01% high)

For √218 the tangent at 15 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 218 is just 7 below 225.

Finding √218 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, 15 (15² = 225):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √218 = 14.7648230602 to every decimal shown.

√218 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √218 the pattern is [14; 1, 3, 3, 1, 28] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √218 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 − 218y² = 1. Its smallest solution in positive whole numbers is x = 126,003, y = 8,534. Because the period is odd, the equation with −1 on the right also has a solution: 251² − 218 × 17² = −1.

√218 in geometry and everyday measurements

• A square patio or deck of 218 square feet is about 14.76 ft (14 ft 9 in) on each side, so edging all the way around takes 4 × √218 ≈ 59.1 ft.
• 218 = 7² + 13², so by the Pythagorean theorem √218 is the diagonal of a 7 × 13 rectangle — and the distance between the points (0, 0) and (7, 13) on a grid.

Square roots near √218 and related roots

• The cube root of 218 is about 6.018462.
• Four times the radicand doubles the root: √872 = 2 × √218 ≈ 29.529646.

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