Square Root of 227

√227 at a glance

Exact value

√227

Decimal (10 places)

15.0665191733

Rounded

15.1 · 15.07 · 15.067

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.066519

Prime factorization

227

Cube root

6.100170

How to simplify √227

227 is a prime number, so its only factors are 1 and 227. There is no perfect-square factor to pull out, which means √227 is already in its simplest radical form.

The square root of any prime is irrational. If √227 were a fraction a/b in lowest terms, then a² = 227b², so 227 would divide a — and then 227 would divide b too, contradicting “lowest terms.” That is why the decimal 15.0665191733 is only a rounded value.

Where √227 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √227 lies between 15 and 16. 227 is 2 above 225 and 29 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.0645 (0.01% low)
• Tangent from 15, i.e. 15 + 2 ÷ 30: 15.0667 (0% high)
• Tangent from 16, i.e. 16 − 29 ÷ 32: 15.0938 (0.18% high)

For √227 the tangent at 15 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 227 is just 2 above 225.

Finding √227 with the Babylonian method

If a guess is too big, 227 ÷ 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√227) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √227 = 15.0665191733 to every decimal shown.

√227 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √227 the pattern is [15; 15, 30] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √227 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 − 227y² = 1. Its smallest solution in positive whole numbers is x = 226, y = 15.

√227 in geometry and everyday measurements

• A square patio or deck of 227 square feet is about 15.07 ft (15 ft 1 in) on each side, so edging all the way around takes 4 × √227 ≈ 60.3 ft.
• 227 is not a sum of two whole-number squares — 227 is itself a prime that is one less than a multiple of 4, which rules that out — so √227 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 15 box, because 1² + 1² + 15² = 227.

Square roots near √227 and related roots

• The cube root of 227 is about 6.100170.
• Four times the radicand doubles the root: √908 = 2 × √227 ≈ 30.133038.

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