√366 at a glance
- Exact value
- √366
- Decimal (10 places)
- 19.1311264697
- Rounded
- 19.1 · 19.13 · 19.131
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.131126
- Prime factorization
- 2 × 3 × 61
- Cube root
- 7.153090
How to simplify √366
The prime factorization of 366 is 2 × 3 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √366 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 366, 2, 3 and 61 appear an odd number of times, so √366 is irrational and 19.1311264697 is a rounded value.
Where √366 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √366 lies between 19 and 20. 366 is 5 above 361 and 34 below 400, so the root is closer to 19.
- Straight line between 361 and 400: 19.1282 (0.02% low)
- Tangent from 19, i.e. 19 + 5 ÷ 38: 19.1316 (0% high)
- Tangent from 20, i.e. 20 − 34 ÷ 40: 19.1500 (0.1% high)
For √366 the tangent at 19 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 366 is just 5 above 361.
Finding √366 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, 19 (19² = 361):
| Step | Guess x | 366 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 19.2631578947 | 19.1315789474 | 3 |
| 2 | 19.1315789474 | 19.1306740028 | 19.1311264751 | 8 |
| 3 | 19.1311264751 | 19.1311264644 | 19.1311264697 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √366 = 19.1311264697 to every decimal shown.
√366 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √366 the pattern is [19; 7, 1, 1, 1, 2, 12, 2, 1, 1, 1, 7, 38] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √366 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 |
|---|---|---|
| 19/1 | 19.0000000000 | 1.3 × 10⁻¹ |
| 134/7 | 19.1428571429 | 1.2 × 10⁻² |
| 153/8 | 19.1250000000 | 6.1 × 10⁻³ |
| 287/15 | 19.1333333333 | 2.2 × 10⁻³ |
| 440/23 | 19.1304347826 | 6.9 × 10⁻⁴ |
| 1,167/61 | 19.1311475410 | 2.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 366y² = 1. Its smallest solution in positive whole numbers is x = 907,925, y = 47,458.
√366 in geometry and everyday measurements
- A square patio or deck of 366 square feet is about 19.13 ft (19 ft 2 in) on each side, so edging all the way around takes 4 × √366 ≈ 76.5 ft.
- 366 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 √366 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 19 box, because 1² + 2² + 19² = 366.
Square roots near √366 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √363 | 11√3 | 19.0526 | No |
| √364 | 2√91 | 19.0788 | No |
| √365 | √365 | 19.1050 | No |
| √366 | √366 | 19.1311 | No |
| √367 | √367 | 19.1572 | No |
| √368 | 4√23 | 19.1833 | No |
| √369 | 3√41 | 19.2094 | No |
- The cube root of 366 is about 7.153090.
- Squaring undoes the root: (√366)² = 366, while 366² = 133,956 — the number whose square root is 366.
Frequently asked questions
What is the square root of 366?
The square root of 366 is √366, about 19.1311264697. The negative root, −19.131126, also squares to 366.
Is the square root of 366 rational or irrational?
Irrational. 366 is not a perfect square — it falls between 361 and 400 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √366 be simplified?
No. 366 = 2 × 3 × 61 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √366 rounded to two decimal places?
√366 ≈ 19.13 to two decimal places (19.1 to one, 19.131 to three). Check: 19.13² = 365.9569, close to 366.