Square Root of 210

√210 at a glance

Exact value

√210

Decimal (10 places)

14.4913767462

Rounded

14.5 · 14.49 · 14.491

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.491377

Prime factorization

2 × 3 × 5 × 7

Cube root

5.943922

How to simplify √210

The prime factorization of 210 is 2 × 3 × 5 × 7. Every prime appears only once, so there is no pair to bring outside the radical — √210 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 210, 2, 3, 5 and 7 appear an odd number of times, so √210 is irrational and 14.4913767462 is a rounded value.

Where √210 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √210 lies between 14 and 15. 210 is 14 above 196 and 15 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.4828 (0.06% low)
• Tangent from 14, i.e. 14 + 14 ÷ 28: 14.5000 (0.06% high)
• Tangent from 15, i.e. 15 − 15 ÷ 30: 14.5000 (0.06% high)

For √210 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √210 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, 14 (14² = 196):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √210 = 14.4913767462 to every decimal shown.

√210 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √210 the pattern is [14; 2, 28] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √210 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 − 210y² = 1. Its smallest solution in positive whole numbers is x = 29, y = 2.

√210 in geometry and everyday measurements

• A square patio or deck of 210 square feet is about 14.49 ft (14 ft 6 in) on each side, so edging all the way around takes 4 × √210 ≈ 58 ft.
• 210 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √210 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 5 × 13 box, because 4² + 5² + 13² = 210.

Square roots near √210 and related roots

• The cube root of 210 is about 5.943922.
• Four times the radicand doubles the root: √840 = 2 × √210 ≈ 28.982753.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.