Square Root of 162

√162 at a glance

Exact value

9√2

Decimal (10 places)

12.7279220614

Rounded

12.7 · 12.73 · 12.728

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.727922

Prime factorization

2 × 3⁴

Cube root

5.451362

How to simplify √162

Look for the largest perfect square that divides 162. Here it is 81 (9²), because 162 = 81 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 162 = 2 × 3⁴. Each pair of equal primes leaves the radical as one factor, so 3² comes out and 2 stays inside.

162 has 2 square factors (9 and 81). Starting with a smaller one still works but takes more rounds: √162 = 3√18, and √18 can be simplified again. Using 81 straight away finishes in one step.

Check: (9√2)² = 9² × 2 = 81 × 2 = 162. As a decimal, 9√2 = 9 × 1.4142135624 ≈ 12.7279220614.

Where √162 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √162 lies between 12 and 13. 162 is 18 above 144 and 7 below 169, so the root is closer to 13.

• Straight line between 144 and 169: 12.7200 (0.06% low)
• Tangent from 12, i.e. 12 + 18 ÷ 24: 12.7500 (0.17% high)
• Tangent from 13, i.e. 13 − 7 ÷ 26: 12.7308 (0.02% high)

For √162 the tangent at 13 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 162 is just 7 below 169.

Finding √162 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, 13 (13² = 169):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √162 = 12.7279220614 to every decimal shown.

√162 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √162 the pattern is [12; 1, 2, 1, 2, 12, 2, 1, 2, 1, 24] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √162 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 − 162y² = 1. Its smallest solution in positive whole numbers is x = 19,601, y = 1,540.

√162 in geometry and everyday measurements

• A square room or garden bed covering 162 square feet measures about 12.73 ft (12 ft 9 in) along each wall.
• 162 = 9² + 9², so by the Pythagorean theorem √162 is the diagonal of a 9 × 9 rectangle — and the distance between the points (0, 0) and (9, 9) on a grid.
• Since √162 = 9√2, a length of √162 is exactly 9 copies of the length √2 laid end to end.

Square roots near √162 and related roots

• The cube root of 162 is about 5.451362.
• Four times the radicand doubles the root: √648 = 2 × √162 ≈ 25.455844.

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