√361 at a glance
- Exact value
- 19
- Decimal (10 places)
- 19
- Rounded
- 19
- Perfect square?
- Yes — 19²
- Rational?
- Rational (whole number)
- Both square roots
- ±19
- Prime factorization
- 19²
- Cube root
- 7.120367
How to simplify √361
Write 361 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√361 = 19 = 19
361 has 2 factor pairs (1 × 361, 19 × 19). The pair with two equal numbers, 19 × 19, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 361 = 1 + 3 + 5 + 7 + … + 37, the first 19 odd numbers.
Where √361 sits between perfect squares
361 is 19². The perfect squares on either side are 324 (18²) and 400 (20²), so the gaps are 37 below and 39 above — consecutive squares always differ by consecutive odd numbers.
Finding √361 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 361: following the tangent line down to zero simplifies to averaging x with 361 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 20:
| Step | Guess x | 361 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 18.0500000000 | 19.0250000000 | 1 |
| 2 | 19.0250000000 | 18.9750328515 | 19.0000164258 | 4 |
| 3 | 19.0000164258 | 18.9999835743 | 19.0000000000 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √361 = 19.0000000000 to every decimal shown.
√361 in geometry and everyday measurements
- A square patio or deck of 361 square feet is exactly 19 ft (19 ft) on each side, so edging all the way around takes 4 × √361 ≈ 76 ft.
- 361 is not a sum of two whole-number squares — no pair of squares adds up to it — so √361 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 18 box, because 1² + 6² + 18² = 361.
Square roots near √361 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √358 | √358 | 18.9209 | No |
| √359 | √359 | 18.9473 | No |
| √360 | 6√10 | 18.9737 | No |
| √361 | 19 | 19.0000 | Yes |
| √362 | √362 | 19.0263 | No |
| √363 | 11√3 | 19.0526 | No |
| √364 | 2√91 | 19.0788 | No |
- The cube root of 361 is about 7.120367.
- Taking the square root twice gives the fourth root: √19 ≈ 4.358899.
- Squaring undoes the root: (√361)² = 361, while 361² = 130,321 — the number whose square root is 361.
Frequently asked questions
What is the square root of 361?
The square root of 361 is 19, because 19 × 19 = 361. Strictly, 361 has two square roots, 19 and −19; the √ symbol means the positive one.
Is the square root of 361 rational or irrational?
Rational. 361 is a perfect square, so its square root is the whole number 19.
Can √361 be simplified?
Yes, all the way to a whole number: √361 = 19.
What is √361 rounded to two decimal places?
√361 is exactly 19, or 19.00 written to two decimal places.