Square Root of 221

√221 at a glance

Exact value

√221

Decimal (10 places)

14.8660687473

Rounded

14.9 · 14.87 · 14.866

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.866069

Prime factorization

13 × 17

Cube root

6.045944

How to simplify √221

The prime factorization of 221 is 13 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √221 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 221, 13 and 17 appear an odd number of times, so √221 is irrational and 14.8660687473 is a rounded value.

Where √221 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √221 lies between 14 and 15. 221 is 25 above 196 and 4 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.8621 (0.03% low)
• Tangent from 14, i.e. 14 + 25 ÷ 28: 14.8929 (0.18% high)
• Tangent from 15, i.e. 15 − 4 ÷ 30: 14.8667 (0% high)

For √221 the tangent at 15 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 221 is just 4 below 225.

Finding √221 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 221: following the tangent line down to zero simplifies to averaging x with 221 ÷ x.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √221 = 14.8660687473 to every decimal shown.

√221 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √221 the pattern is [14; 1, 6, 2, 6, 1, 28] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √221 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 − 221y² = 1. Its smallest solution in positive whole numbers is x = 1,665, y = 112.

√221 in geometry and everyday measurements

• A square patio or deck of 221 square feet is about 14.87 ft (14 ft 10 in) on each side, so edging all the way around takes 4 × √221 ≈ 59.5 ft.
• 221 = 5² + 14² = 10² + 11², so by the Pythagorean theorem √221 is the diagonal of rectangles measuring 5 × 14 and 10 × 11 — and the distance between the points (0, 0) and (5, 14) on a grid.

Square roots near √221 and related roots

• The cube root of 221 is about 6.045944.
• Four times the radicand doubles the root: √884 = 2 × √221 ≈ 29.732137.

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