Square Root of 122

√122 at a glance

Exact value

√122

Decimal (10 places)

11.0453610172

Rounded

11.0 · 11.05 · 11.045

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.045361

Prime factorization

2 × 61

Cube root

4.959676

How to simplify √122

The prime factorization of 122 is 2 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √122 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 122, 2 and 61 appear an odd number of times, so √122 is irrational and 11.0453610172 is a rounded value.

Where √122 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √122 lies between 11 and 12. 122 is 1 above 121 and 22 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.0435 (0.02% low)
• Tangent from 11, i.e. 11 + 1 ÷ 22: 11.0455 (0% high)
• Tangent from 12, i.e. 12 − 22 ÷ 24: 11.0833 (0.34% high)

For √122 the tangent at 11 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 122 is just 1 above 121.

Finding √122 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, 11 (11² = 121):

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √122 = 11.0453610172 to every decimal shown.

√122 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √122 the pattern is [11; 22] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 122 is one more than a perfect square (11² + 1). A pattern that never ends is one more proof that √122 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 − 122y² = 1. Its smallest solution in positive whole numbers is x = 243, y = 22. Because the period is odd, the equation with −1 on the right also has a solution: 11² − 122 × 1² = −1.

√122 in geometry and everyday measurements

• A square room or garden bed covering 122 square feet measures about 11.05 ft (11 ft 1 in) along each wall.
• 122 = 1² + 11², so by the Pythagorean theorem √122 is the diagonal of a 1 × 11 rectangle — and the distance between the points (0, 0) and (1, 11) on a grid.

Square roots near √122 and related roots

• The cube root of 122 is about 4.959676.
• Four times the radicand doubles the root: √488 = 2 × √122 ≈ 22.090722.

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