Square Root of 118

√118 at a glance

Exact value

√118

Decimal (10 places)

10.8627804912

Rounded

10.9 · 10.86 · 10.863

Perfect square?

No — between 10² and 11²

Rational?

Irrational

Both square roots

±10.862780

Prime factorization

2 × 59

Cube root

4.904868

How to simplify √118

The prime factorization of 118 is 2 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √118 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 118, 2 and 59 appear an odd number of times, so √118 is irrational and 10.8627804912 is a rounded value.

Where √118 sits between perfect squares

100 = 10² and 121 = 11² are the nearest perfect squares, so √118 lies between 10 and 11. 118 is 18 above 100 and 3 below 121, so the root is closer to 11.

• Straight line between 100 and 121: 10.8571 (0.05% low)
• Tangent from 10, i.e. 10 + 18 ÷ 20: 10.9000 (0.34% high)
• Tangent from 11, i.e. 11 − 3 ÷ 22: 10.8636 (0.01% high)

For √118 the tangent at 11 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 118 is just 3 below 121.

Finding √118 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 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √118 = 10.8627804912 to every decimal shown.

√118 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √118 the pattern is [10; 1, 6, 3, 2, 10, 2, 3, 6, 1, 20] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √118 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 − 118y² = 1. Its smallest solution in positive whole numbers is x = 306,917, y = 28,254.

√118 in geometry and everyday measurements

• A square room or garden bed covering 118 square feet measures about 10.86 ft (10 ft 10 in) along each wall.
• 118 is not a sum of two whole-number squares — the prime factor 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √118 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 9 box, because 1² + 6² + 9² = 118.

Square roots near √118 and related roots

• The cube root of 118 is about 4.904868.
• Four times the radicand doubles the root: √472 = 2 × √118 ≈ 21.725561.

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