√330 at a glance
- Exact value
- √330
- Decimal (10 places)
- 18.1659021246
- Rounded
- 18.2 · 18.17 · 18.166
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.165902
- Prime factorization
- 2 × 3 × 5 × 11
- Cube root
- 6.910423
How to simplify √330
The prime factorization of 330 is 2 × 3 × 5 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √330 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 330, 2, 3, 5 and 11 appear an odd number of times, so √330 is irrational and 18.1659021246 is a rounded value.
Where √330 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √330 lies between 18 and 19. 330 is 6 above 324 and 31 below 361, so the root is closer to 18.
- Straight line between 324 and 361: 18.1622 (0.02% low)
- Tangent from 18, i.e. 18 + 6 ÷ 36: 18.1667 (0% high)
- Tangent from 19, i.e. 19 − 31 ÷ 38: 18.1842 (0.1% high)
For √330 the tangent at 18 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 330 is just 6 above 324.
Finding √330 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, 18 (18² = 324):
| Step | Guess x | 330 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 18.0000000000 | 18.3333333333 | 18.1666666667 | 3 |
| 2 | 18.1666666667 | 18.1651376147 | 18.1659021407 | 7 |
| 3 | 18.1659021407 | 18.1659021085 | 18.1659021246 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √330 = 18.1659021246 to every decimal shown.
√330 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √330 the pattern is [18; 6, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √330 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 18/1 | 18.0000000000 | 1.7 × 10⁻¹ |
| 109/6 | 18.1666666667 | 7.6 × 10⁻⁴ |
| 3,942/217 | 18.1658986175 | 3.5 × 10⁻⁶ |
| 23,761/1,308 | 18.1659021407 | 1.6 × 10⁻⁸ |
| 859,338/47,305 | 18.1659021245 | 7.4 × 10⁻¹¹ |
| 5,179,789/285,138 | 18.1659021246 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 330y² = 1. Its smallest solution in positive whole numbers is x = 109, y = 6.
√330 in geometry and everyday measurements
- A square patio or deck of 330 square feet is about 18.17 ft (18 ft 2 in) on each side, so edging all the way around takes 4 × √330 ≈ 72.7 ft.
- 330 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √330 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 5 × 17 box, because 4² + 5² + 17² = 330.
Square roots near √330 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √327 | √327 | 18.0831 | No |
| √328 | 2√82 | 18.1108 | No |
| √329 | √329 | 18.1384 | No |
| √330 | √330 | 18.1659 | No |
| √331 | √331 | 18.1934 | No |
| √332 | 2√83 | 18.2209 | No |
| √333 | 3√37 | 18.2483 | No |
- The cube root of 330 is about 6.910423.
- Squaring undoes the root: (√330)² = 330, while 330² = 108,900 — the number whose square root is 330.
Frequently asked questions
What is the square root of 330?
The square root of 330 is √330, about 18.1659021246. The negative root, −18.165902, also squares to 330.
Is the square root of 330 rational or irrational?
Irrational. 330 is not a perfect square — it falls between 324 and 361 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √330 be simplified?
No. 330 = 2 × 3 × 5 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √330 rounded to two decimal places?
√330 ≈ 18.17 to two decimal places (18.2 to one, 18.166 to three). Check: 18.17² = 330.1489, close to 330.