Square Root Calculator › Square root of 309
Square Root of 309
√309 ≈ 17.578396
309 is not a perfect square, so √309 is an irrational number that is approximately 17.578396. In simplified radical form, √309 = √309.
√309 lies between √289 = 17 and √324 = 18.
| Number | 309 |
| Square root (decimal) | 17.578396 |
| Simplified radical | √309 |
| Perfect square? | No |
| Rational? | No |
Calculate another square root
Prime factorization of 309
309 = 3 × 103. This is what determines the simplified radical: for √309 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √309 is already in simplest radical form as √309.
Estimating √309 by hand
√309 falls between √289 = 17 and √324 = 18, so it starts at 17.something. The Babylonian method refines that guess by averaging it with 309 ÷ the guess:
| Step | Estimate |
| Start | 17 |
| Step 1 | 17.588235 |
| Step 2 | 17.578399 |
| Step 3 | 17.578396 |
After 3 steps the estimate is 17.578396, matching √309 ≈ 17.578396.
Nearby square roots
Frequently asked questions
Is 309 a perfect square?
No, 309 is not a perfect square, so its square root is irrational and approximately 17.578396.
What is the square root of 309 in simplest radical form?
√309 = √309 in simplest radical form.