Square Root of 12
The square root of 12 is approximately 3.464102, or exactly 2√3 in simplified radical form. No, 12 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √12 sits
3² = 9 and 4² = 16, and 12 falls between them, so √12 is between 3 and 4. It's closer to 3 (3 above 9, 4 below 16), which matches the decimal 3.464102.
| k | k² | Compared to 12 |
|---|---|---|
| 1 | 1 | 11 below |
| 2 | 4 | 8 below |
| 3 | 9 | 3 below |
| 4 | 16 | 4 above |
| 5 | 25 | 13 above |
| 6 | 36 | 24 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 3), divide 12 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 12 ÷ x | New guess (x + 12÷x) ÷ 2 |
|---|---|---|---|
| 1 | 3 | 4 | 3.5 |
| 2 | 3.5 | 3.428571 | 3.464286 |
| 3 | 3.464286 | 3.463918 | 3.464102 |
| 4 | 3.464102 | 3.464102 | 3.464102 |
After 4 passes the guess is 3.464102, already matching √12 = 3.464102 to six decimal places.
Exact form from the prime factors
12 = 2^2 × 3. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √12 = 2√3. That's the exact value; the decimal 3.464102 is a rounding of it.
Frequently Asked Questions
What is the square root of 12?
The square root of 12 is approximately 3.464102, or exactly 2√3 in simplified radical form.
Is 12 a perfect square?
No, 12 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √12 between?
Between 3 and 4, because 3² = 9 and 4² = 16.
What are the square roots of the numbers right next to 12?
√11 is about 3.3166, and √13 is about 3.6056. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √12 squared?
Exactly 12 -- squaring undoes the square root. Squaring the rounded decimal 3.464102 gives 12, which is why the radical form 2√3 is the exact answer and the decimal is an approximation.
What is the negative square root of 12?
−3.464102. Every positive number has two square roots, one positive and one negative, because (−3.464102)² is also 12. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
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