Square Root Calculator › Square root of 462
Square Root of 462
√462 ≈ 21.494185
462 is not a perfect square, so √462 is an irrational number that is approximately 21.494185. In simplified radical form, √462 = √462.
√462 lies between √441 = 21 and √484 = 22.
| Number | 462 |
| Square root (decimal) | 21.494185 |
| Simplified radical | √462 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 462
462 = 2 × 3 × 7 × 11. This is what determines the simplified radical: for √462 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √462 is already in simplest radical form as √462.
Estimating √462 by hand
√462 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 462 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.5 |
| Step 2 | 21.494186 |
| Step 3 | 21.494185 |
After 3 steps the estimate is 21.494185, matching √462 ≈ 21.494185.
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Frequently asked questions
Is 462 a perfect square?
No, 462 is not a perfect square, so its square root is irrational and approximately 21.494185.
What is the square root of 462 in simplest radical form?
√462 = √462 in simplest radical form.