Square Root of 226

√226 at a glance

Exact value

√226

Decimal (10 places)

15.0332963784

Rounded

15.0 · 15.03 · 15.033

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.033296

Prime factorization

2 × 113

Cube root

6.091199

How to simplify √226

The prime factorization of 226 is 2 × 113. Every prime appears only once, so there is no pair to bring outside the radical — √226 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 226, 2 and 113 appear an odd number of times, so √226 is irrational and 15.0332963784 is a rounded value.

Where √226 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √226 lies between 15 and 16. 226 is 1 above 225 and 30 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.0323 (0.01% low)
• Tangent from 15, i.e. 15 + 1 ÷ 30: 15.0333 (0% high)
• Tangent from 16, i.e. 16 − 30 ÷ 32: 15.0625 (0.19% high)

For √226 the tangent at 15 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 226 is just 1 above 225.

Finding √226 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

Because the starting guess was already close, two steps are enough to match √226 = 15.0332963784 to every decimal shown.

√226 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √226 the pattern is [15; 30] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 226 is one more than a perfect square (15² + 1). A pattern that never ends is one more proof that √226 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 − 226y² = 1. Its smallest solution in positive whole numbers is x = 451, y = 30. Because the period is odd, the equation with −1 on the right also has a solution: 15² − 226 × 1² = −1.

√226 in geometry and everyday measurements

• A square patio or deck of 226 square feet is about 15.03 ft (15 ft) on each side, so edging all the way around takes 4 × √226 ≈ 60.1 ft.
• 226 = 1² + 15², so by the Pythagorean theorem √226 is the diagonal of a 1 × 15 rectangle — and the distance between the points (0, 0) and (1, 15) on a grid.

Square roots near √226 and related roots

• The cube root of 226 is about 6.091199.
• Four times the radicand doubles the root: √904 = 2 × √226 ≈ 30.066593.

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