Square Root Calculator › Square root of 312
Square Root of 312
√312 ≈ 17.663522
312 is not a perfect square, so √312 is an irrational number that is approximately 17.663522. In simplified radical form, √312 = 2√78.
√312 lies between √289 = 17 and √324 = 18.
| Number | 312 |
| Square root (decimal) | 17.663522 |
| Simplified radical | 2√78 |
| Perfect square? | No |
| Rational? | No |
Calculate another square root
Prime factorization of 312
312 = 2^3 × 3 × 13. This is what determines the simplified radical: for √312 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √312 = 2√78.
Estimating √312 by hand
√312 falls between √289 = 17 and √324 = 18, so it starts at 17.something. The Babylonian method refines that guess by averaging it with 312 ÷ the guess:
| Step | Estimate |
| Start | 17 |
| Step 1 | 17.676471 |
| Step 2 | 17.663526 |
| Step 3 | 17.663522 |
After 3 steps the estimate is 17.663522, matching √312 ≈ 17.663522.
Nearby square roots
Frequently asked questions
Is 312 a perfect square?
No, 312 is not a perfect square, so its square root is irrational and approximately 17.663522.
What is the square root of 312 in simplest radical form?
√312 = 2√78 in simplest radical form.