Square Root of 229

√229 at a glance

Exact value

√229

Decimal (10 places)

15.1327459504

Rounded

15.1 · 15.13 · 15.133

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.132746

Prime factorization

229

Cube root

6.118033

How to simplify √229

229 is a prime number, so its only factors are 1 and 229. There is no perfect-square factor to pull out, which means √229 is already in its simplest radical form.

The square root of any prime is irrational. If √229 were a fraction a/b in lowest terms, then a² = 229b², so 229 would divide a — and then 229 would divide b too, contradicting “lowest terms.” That is why the decimal 15.1327459504 is only a rounded value.

Where √229 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √229 lies between 15 and 16. 229 is 4 above 225 and 27 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.1290 (0.02% low)
• Tangent from 15, i.e. 15 + 4 ÷ 30: 15.1333 (0% high)
• Tangent from 16, i.e. 16 − 27 ÷ 32: 15.1563 (0.16% high)

For √229 the tangent at 15 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 229 is just 4 above 225.

Finding √229 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 229: following the tangent line down to zero simplifies to averaging x with 229 ÷ 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 √229 = 15.1327459504 to every decimal shown.

√229 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √229 the pattern is [15; 7, 1, 1, 7, 30] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √229 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 − 229y² = 1. Its smallest solution in positive whole numbers is x = 5,848,201, y = 386,460. Because the period is odd, the equation with −1 on the right also has a solution: 1,710² − 229 × 113² = −1.

√229 in geometry and everyday measurements

• A square patio or deck of 229 square feet is about 15.13 ft (15 ft 2 in) on each side, so edging all the way around takes 4 × √229 ≈ 60.5 ft.
• 229 = 2² + 15², so by the Pythagorean theorem √229 is the diagonal of a 2 × 15 rectangle — and the distance between the points (0, 0) and (2, 15) on a grid.

Square roots near √229 and related roots

• The cube root of 229 is about 6.118033.
• Four times the radicand doubles the root: √916 = 2 × √229 ≈ 30.265492.

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