Square Root of 213

√213 at a glance

Exact value

√213

Decimal (10 places)

14.5945195193

Rounded

14.6 · 14.59 · 14.595

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.594520

Prime factorization

3 × 71

Cube root

5.972093

How to simplify √213

The prime factorization of 213 is 3 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √213 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 213, 3 and 71 appear an odd number of times, so √213 is irrational and 14.5945195193 is a rounded value.

Where √213 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √213 lies between 14 and 15. 213 is 17 above 196 and 12 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.5862 (0.06% low)
• Tangent from 14, i.e. 14 + 17 ÷ 28: 14.6071 (0.09% high)
• Tangent from 15, i.e. 15 − 12 ÷ 30: 14.6000 (0.04% high)

For √213 the tangent at 15 wins, missing by only 0.0055. Tangent estimates shine when the number sits close to a perfect square — here 213 is just 12 below 225.

Finding √213 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 213: following the tangent line down to zero simplifies to averaging x with 213 ÷ x.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √213 = 14.5945195193 to every decimal shown.

√213 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √213 the pattern is [14; 1, 1, 2, 6, 1, 8, 1, 6, 2, 1, 1, 28] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √213 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 − 213y² = 1. Its smallest solution in positive whole numbers is x = 194,399, y = 13,320.

√213 in geometry and everyday measurements

• A square patio or deck of 213 square feet is about 14.59 ft (14 ft 7 in) on each side, so edging all the way around takes 4 × √213 ≈ 58.4 ft.
• 213 is not a sum of two whole-number squares — the prime factor 3 and 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √213 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 14 box, because 1² + 4² + 14² = 213.

Square roots near √213 and related roots

• The cube root of 213 is about 5.972093.
• Four times the radicand doubles the root: √852 = 2 × √213 ≈ 29.189039.

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