Square Root of 18

√18 at a glance

Exact value

3√2

Decimal (10 places)

4.2426406871

Rounded

4.2 · 4.24 · 4.243

Perfect square?

No — between 4² and 5²

Rational?

Irrational

Both square roots

±4.242641

Prime factorization

2 × 3²

Cube root

2.620741

How to simplify √18

Look for the largest perfect square that divides 18. Here it is 9 (3²), because 18 = 9 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 18 = 2 × 3². Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 stays inside.

Check: (3√2)² = 3² × 2 = 9 × 2 = 18. As a decimal, 3√2 = 3 × 1.4142135624 ≈ 4.2426406871.

Where √18 sits between perfect squares

16 = 4² and 25 = 5² are the nearest perfect squares, so √18 lies between 4 and 5. 18 is 2 above 16 and 7 below 25, so the root is closer to 4.

• Straight line between 16 and 25: 4.2222 (0.48% low)
• Tangent from 4, i.e. 4 + 2 ÷ 8: 4.2500 (0.17% high)
• Tangent from 5, i.e. 5 − 7 ÷ 10: 4.3000 (1.35% high)

For √18 the tangent at 4 wins, missing by only 0.0074. Tangent estimates shine when the number sits close to a perfect square — here 18 is just 2 above 16.

Finding √18 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, 4 (4² = 16):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √18 = 4.2426406871 to every decimal shown.

√18 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √18 the pattern is [4; 4, 8] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √18 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 − 18y² = 1. Its smallest solution in positive whole numbers is x = 17, y = 4.

√18 in geometry and everyday measurements

• A square tile with an area of 18 square inches has sides about 4.243 in long.
• 18 = 3² + 3², so by the Pythagorean theorem √18 is the diagonal of a 3 × 3 rectangle — and the distance between the points (0, 0) and (3, 3) on a grid.
• Since √18 = 3√2, a length of √18 is exactly 3 copies of the length √2 laid end to end.

Square roots near √18 and related roots

• The cube root of 18 is about 2.620741.
• Four times the radicand doubles the root: √72 = 2 × √18 ≈ 8.485281.

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