√367 at a glance
- Exact value
- √367
- Decimal (10 places)
- 19.1572440607
- Rounded
- 19.2 · 19.16 · 19.157
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.157244
- Prime factorization
- 367
- Cube root
- 7.159599
How to simplify √367
367 is a prime number, so its only factors are 1 and 367. There is no perfect-square factor to pull out, which means √367 is already in its simplest radical form.
The square root of any prime is irrational. If √367 were a fraction a/b in lowest terms, then a² = 367b², so 367 would divide a — and then 367 would divide b too, contradicting “lowest terms.” That is why the decimal 19.1572440607 is only a rounded value.
Where √367 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √367 lies between 19 and 20. 367 is 6 above 361 and 33 below 400, so the root is closer to 19.
- Straight line between 361 and 400: 19.1538 (0.02% low)
- Tangent from 19, i.e. 19 + 6 ÷ 38: 19.1579 (0% high)
- Tangent from 20, i.e. 20 − 33 ÷ 40: 19.1750 (0.09% high)
For √367 the tangent at 19 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 367 is just 6 above 361.
Finding √367 with the Babylonian method
If a guess is too big, 367 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√367) in one step.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 367 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 19.3157894737 | 19.1578947368 | 3 |
| 2 | 19.1578947368 | 19.1565934066 | 19.1572440717 | 7 |
| 3 | 19.1572440717 | 19.1572440496 | 19.1572440607 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √367 = 19.1572440607 to every decimal shown.
√367 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √367 the pattern is [19; 6, 2, 1, 3, 1, 1, 2, 1, 12, 19, 12, 1, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √367 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 19/1 | 19.0000000000 | 1.6 × 10⁻¹ |
| 115/6 | 19.1666666667 | 9.4 × 10⁻³ |
| 249/13 | 19.1538461538 | 3.4 × 10⁻³ |
| 364/19 | 19.1578947368 | 6.5 × 10⁻⁴ |
| 1,341/70 | 19.1571428571 | 1.0 × 10⁻⁴ |
| 1,705/89 | 19.1573033708 | 5.9 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 367y² = 1. Its smallest solution in positive whole numbers is x = 19,019,995,568, y = 992,835,687.
√367 in geometry and everyday measurements
- A square patio or deck of 367 square feet is about 19.16 ft (19 ft 2 in) on each side, so edging all the way around takes 4 × √367 ≈ 76.6 ft.
- 367 is not a sum of two whole-number squares — 367 is itself a prime that is one less than a multiple of 4, which rules that out — so √367 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √367 as its space diagonal.
Square roots near √367 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √364 | 2√91 | 19.0788 | No |
| √365 | √365 | 19.1050 | No |
| √366 | √366 | 19.1311 | No |
| √367 | √367 | 19.1572 | No |
| √368 | 4√23 | 19.1833 | No |
| √369 | 3√41 | 19.2094 | No |
| √370 | √370 | 19.2354 | No |
- The cube root of 367 is about 7.159599.
- Squaring undoes the root: (√367)² = 367, while 367² = 134,689 — the number whose square root is 367.
Frequently asked questions
What is the square root of 367?
The square root of 367 is √367, about 19.1572440607. The negative root, −19.157244, also squares to 367.
Is the square root of 367 rational or irrational?
Irrational. 367 is not a perfect square — it falls between 361 and 400 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √367 be simplified?
No. 367 is prime, so there is no perfect square to take out of the radical.
What is √367 rounded to two decimal places?
√367 ≈ 19.16 to two decimal places (19.2 to one, 19.157 to three). Check: 19.16² = 367.1056, close to 367.