Square Root of 150

√150 at a glance

Exact value

5√6

Decimal (10 places)

12.2474487139

Rounded

12.2 · 12.25 · 12.247

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.247449

Prime factorization

2 × 3 × 5²

Cube root

5.313293

How to simplify √150

Look for the largest perfect square that divides 150. Here it is 25 (5²), because 150 = 25 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 150 = 2 × 3 × 5². Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 3 stays inside.

Check: (5√6)² = 5² × 6 = 25 × 6 = 150. As a decimal, 5√6 = 5 × 2.4494897428 ≈ 12.2474487139.

Where √150 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √150 lies between 12 and 13. 150 is 6 above 144 and 19 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.2400 (0.06% low)
• Tangent from 12, i.e. 12 + 6 ÷ 24: 12.2500 (0.02% high)
• Tangent from 13, i.e. 13 − 19 ÷ 26: 12.2692 (0.18% high)

For √150 the tangent at 12 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 150 is just 6 above 144.

Finding √150 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, 12 (12² = 144):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √150 = 12.2474487139 to every decimal shown.

√150 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √150 the pattern is [12; 4, 24] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √150 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 − 150y² = 1. Its smallest solution in positive whole numbers is x = 49, y = 4.

√150 in geometry and everyday measurements

• A square room or garden bed covering 150 square feet measures about 12.25 ft (12 ft 3 in) along each wall.
• 150 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 √150 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 10 box, because 1² + 7² + 10² = 150.
• Since √150 = 5√6, a length of √150 is exactly 5 copies of the length √6 laid end to end.

Square roots near √150 and related roots

• The cube root of 150 is about 5.313293.
• Four times the radicand doubles the root: √600 = 2 × √150 ≈ 24.494897.

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