Square Root of 333

√333 at a glance

Exact value

3√37

Decimal (10 places)

18.2482875909

Rounded

18.2 · 18.25 · 18.248

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.248288

Prime factorization

3² × 37

Cube root

6.931301

How to simplify √333

Look for the largest perfect square that divides 333. Here it is 9 (3²), because 333 = 9 × 37 and 37 has no square factor left:

The prime factorization tells the same story: 333 = 3² × 37. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 37 stays inside.

Check: (3√37)² = 3² × 37 = 9 × 37 = 333. As a decimal, 3√37 = 3 × 6.0827625303 ≈ 18.2482875909.

Where √333 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √333 lies between 18 and 19. 333 is 9 above 324 and 28 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.2432 (0.03% low)
• Tangent from 18, i.e. 18 + 9 ÷ 36: 18.2500 (0.01% high)
• Tangent from 19, i.e. 19 − 28 ÷ 38: 18.2632 (0.08% high)

For √333 the tangent at 18 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 333 is just 9 above 324.

Finding √333 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 333: following the tangent line down to zero simplifies to averaging x with 333 ÷ x.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √333 = 18.2482875909 to every decimal shown.

√333 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √333 the pattern is [18; 4, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √333 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 − 333y² = 1. Its smallest solution in positive whole numbers is x = 73, y = 4.

√333 in geometry and everyday measurements

• A square patio or deck of 333 square feet is about 18.25 ft (18 ft 3 in) on each side, so edging all the way around takes 4 × √333 ≈ 73 ft.
• 333 = 3² + 18², so by the Pythagorean theorem √333 is the diagonal of a 3 × 18 rectangle — and the distance between the points (0, 0) and (3, 18) on a grid.
• Since √333 = 3√37, a length of √333 is exactly 3 copies of the length √37 laid end to end.

Square roots near √333 and related roots

• The cube root of 333 is about 6.931301.
• Squaring undoes the root: (√333)² = 333, while 333² = 110,889 — the number whose square root is 333.

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