Square Root of 219

√219 at a glance

Exact value

√219

Decimal (10 places)

14.7986485869

Rounded

14.8 · 14.80 · 14.799

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.798649

Prime factorization

3 × 73

Cube root

6.027650

How to simplify √219

The prime factorization of 219 is 3 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √219 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 219, 3 and 73 appear an odd number of times, so √219 is irrational and 14.7986485869 is a rounded value.

Where √219 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √219 lies between 14 and 15. 219 is 23 above 196 and 6 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.7931 (0.04% low)
• Tangent from 14, i.e. 14 + 23 ÷ 28: 14.8214 (0.15% high)
• Tangent from 15, i.e. 15 − 6 ÷ 30: 14.8000 (0.01% high)

For √219 the tangent at 15 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 219 is just 6 below 225.

Finding √219 with the Babylonian method

If a guess is too big, 219 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√219) in one step.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √219 = 14.7986485869 to every decimal shown.

√219 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √219 the pattern is [14; 1, 3, 1, 28] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √219 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 − 219y² = 1. Its smallest solution in positive whole numbers is x = 74, y = 5.

√219 in geometry and everyday measurements

• A square patio or deck of 219 square feet is about 14.8 ft (14 ft 10 in) on each side, so edging all the way around takes 4 × √219 ≈ 59.2 ft.
• 219 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √219 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 13 box, because 1² + 7² + 13² = 219.

Square roots near √219 and related roots

• The cube root of 219 is about 6.027650.
• Four times the radicand doubles the root: √876 = 2 × √219 ≈ 29.597297.

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