√365 at a glance
- Exact value
- √365
- Decimal (10 places)
- 19.1049731745
- Rounded
- 19.1 · 19.10 · 19.105
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.104973
- Prime factorization
- 5 × 73
- Cube root
- 7.146569
How to simplify √365
The prime factorization of 365 is 5 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √365 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 365, 5 and 73 appear an odd number of times, so √365 is irrational and 19.1049731745 is a rounded value.
Where √365 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √365 lies between 19 and 20. 365 is 4 above 361 and 35 below 400, so the root is closer to 19.
- Straight line between 361 and 400: 19.1026 (0.01% low)
- Tangent from 19, i.e. 19 + 4 ÷ 38: 19.1053 (0% high)
- Tangent from 20, i.e. 20 − 35 ÷ 40: 19.1250 (0.1% high)
For √365 the tangent at 19 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 365 is just 4 above 361.
Finding √365 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 365: following the tangent line down to zero simplifies to averaging x with 365 ÷ x.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 365 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 19.2105263158 | 19.1052631579 | 3 |
| 2 | 19.1052631579 | 19.1046831956 | 19.1049731767 | 8 |
| 3 | 19.1049731767 | 19.1049731723 | 19.1049731745 | 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 √365 = 19.1049731745 to every decimal shown.
√365 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √365 the pattern is [19; 9, 1, 1, 9, 38] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √365 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 | 1.0 × 10⁻¹ |
| 172/9 | 19.1111111111 | 6.1 × 10⁻³ |
| 191/10 | 19.1000000000 | 5.0 × 10⁻³ |
| 363/19 | 19.1052631579 | 2.9 × 10⁻⁴ |
| 3,458/181 | 19.1049723757 | 8.0 × 10⁻⁷ |
| 131,767/6,897 | 19.1049731767 | 2.2 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 365y² = 1. Its smallest solution in positive whole numbers is x = 23,915,529, y = 1,251,796. Because the period is odd, the equation with −1 on the right also has a solution: 3,458² − 365 × 181² = −1.
√365 in geometry and everyday measurements
- A square patio or deck of 365 square feet is about 19.1 ft (19 ft 1 in) on each side, so edging all the way around takes 4 × √365 ≈ 76.4 ft.
- 365 = 2² + 19² = 13² + 14², so by the Pythagorean theorem √365 is the diagonal of rectangles measuring 2 × 19 and 13 × 14 — and the distance between the points (0, 0) and (2, 19) on a grid.
Square roots near √365 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √362 | √362 | 19.0263 | No |
| √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 |
- The cube root of 365 is about 7.146569.
- Squaring undoes the root: (√365)² = 365, while 365² = 133,225 — the number whose square root is 365.
Frequently asked questions
What is the square root of 365?
The square root of 365 is √365, about 19.1049731745. The negative root, −19.104973, also squares to 365.
Is the square root of 365 rational or irrational?
Irrational. 365 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 √365 be simplified?
No. 365 = 5 × 73 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √365 rounded to two decimal places?
√365 ≈ 19.10 to two decimal places (19.1 to one, 19.105 to three). Check: 19.10² = 364.81, close to 365.