Square Root of 120

√120 at a glance

Exact value

2√30

Decimal (10 places)

10.9544511501

Rounded

11.0 · 10.95 · 10.954

Perfect square?

No — between 10² and 11²

Rational?

Irrational

Both square roots

±10.954451

Prime factorization

2³ × 3 × 5

Cube root

4.932424

How to simplify √120

Look for the largest perfect square that divides 120. Here it is 4 (2²), because 120 = 4 × 30 and 30 has no square factor left:

The prime factorization tells the same story: 120 = 2³ × 3 × 5. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 5 stays inside.

Check: (2√30)² = 2² × 30 = 4 × 30 = 120. As a decimal, 2√30 = 2 × 5.4772255751 ≈ 10.9544511501.

Where √120 sits between perfect squares

100 = 10² and 121 = 11² are the nearest perfect squares, so √120 lies between 10 and 11. 120 is 20 above 100 and 1 below 121, so the root is closer to 11.

• Straight line between 100 and 121: 10.9524 (0.02% low)
• Tangent from 10, i.e. 10 + 20 ÷ 20: 11.0000 (0.42% high)
• Tangent from 11, i.e. 11 − 1 ÷ 22: 10.9545 (0% high)

For √120 the tangent at 11 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 120 is just 1 below 121.

Finding √120 with the Babylonian method

Picture a rectangle with an area of 120 and one side x; the other side must be 120 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √120.

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 √120 = 10.9544511501 to every decimal shown.

√120 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √120 the pattern is [10; 1, 20] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √120 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 − 120y² = 1. Its smallest solution in positive whole numbers is x = 11, y = 1.

√120 in geometry and everyday measurements

• A square room or garden bed covering 120 square feet measures about 10.95 ft (10 ft 11 in) along each wall.
• 120 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √120 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 10 box, because 2² + 4² + 10² = 120.
• Since √120 = 2√30, a length of √120 is exactly 2 copies of the length √30 laid end to end.

Square roots near √120 and related roots

• The cube root of 120 is about 4.932424.
• Four times the radicand doubles the root: √480 = 2 × √120 ≈ 21.908902.

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