Square Root Calculator › Square root of 180
Square Root of 180
√180 ≈ 13.416408
180 is not a perfect square, so √180 is an irrational number that is approximately 13.416408. In simplified radical form, √180 = 6√5.
√180 lies between √169 = 13 and √196 = 14.
| Number | 180 |
| Square root (decimal) | 13.416408 |
| Simplified radical | 6√5 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 180
180 = 2^2 × 3^2 × 5. This is what determines the simplified radical: for √180 you pull out one copy of any prime that appears twice.
Here the paired factors 2 and 3 come out from under the root, leaving √180 = 6√5.
Estimating √180 by hand
√180 falls between √169 = 13 and √196 = 14, so it starts at 13.something. The Babylonian method refines that guess by averaging it with 180 ÷ the guess:
| Step | Estimate |
| Start | 13 |
| Step 1 | 13.423077 |
| Step 2 | 13.41641 |
| Step 3 | 13.416408 |
After 3 steps the estimate is 13.416408, matching √180 ≈ 13.416408.
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Frequently asked questions
Is 180 a perfect square?
No, 180 is not a perfect square, so its square root is irrational and approximately 13.416408.
What is the square root of 180 in simplest radical form?
√180 = 6√5 in simplest radical form.