√380 at a glance
- Exact value
- 2√95
- Decimal (10 places)
- 19.4935886896
- Rounded
- 19.5 · 19.49 · 19.494
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.493589
- Prime factorization
- 2² × 5 × 19
- Cube root
- 7.243156
How to simplify √380
Look for the largest perfect square that divides 380. Here it is 4 (2²), because 380 = 4 × 95 and 95 has no square factor left:
The prime factorization tells the same story: 380 = 2² × 5 × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 19 stays inside.
Check: (2√95)² = 2² × 95 = 4 × 95 = 380. As a decimal, 2√95 = 2 × 9.7467943448 ≈ 19.4935886896.
Where √380 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √380 lies between 19 and 20. 380 is 19 above 361 and 20 below 400, so the root is closer to 19.
- Straight line between 361 and 400: 19.4872 (0.03% low)
- Tangent from 19, i.e. 19 + 19 ÷ 38: 19.5000 (0.03% high)
- Tangent from 20, i.e. 20 − 20 ÷ 40: 19.5000 (0.03% high)
For √380 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √380 with the Babylonian method
Picture a rectangle with an area of 380 and one side x; the other side must be 380 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √380.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 380 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 20.0000000000 | 19.5000000000 | 2 |
| 2 | 19.5000000000 | 19.4871794872 | 19.4935897436 | 5 |
| 3 | 19.4935897436 | 19.4935876356 | 19.4935886896 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √380 = 19.4935886896 to every decimal shown.
√380 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √380 the pattern is [19; 2, 38] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √380 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 | 4.9 × 10⁻¹ |
| 39/2 | 19.5000000000 | 6.4 × 10⁻³ |
| 1,501/77 | 19.4935064935 | 8.2 × 10⁻⁵ |
| 3,041/156 | 19.4935897436 | 1.1 × 10⁻⁶ |
| 117,059/6,005 | 19.4935886761 | 1.4 × 10⁻⁸ |
| 237,159/12,166 | 19.4935886898 | 1.7 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 380y² = 1. Its smallest solution in positive whole numbers is x = 39, y = 2.
√380 in geometry and everyday measurements
- A square patio or deck of 380 square feet is about 19.49 ft (19 ft 6 in) on each side, so edging all the way around takes 4 × √380 ≈ 78 ft.
- 380 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √380 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 √380 as its space diagonal.
- Since √380 = 2√95, a length of √380 is exactly 2 copies of the length √95 laid end to end.
Square roots near √380 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √377 | √377 | 19.4165 | No |
| √378 | 3√42 | 19.4422 | No |
| √379 | √379 | 19.4679 | No |
| √380 | 2√95 | 19.4936 | No |
| √381 | √381 | 19.5192 | No |
| √382 | √382 | 19.5448 | No |
| √383 | √383 | 19.5704 | No |
- The cube root of 380 is about 7.243156.
- Because 380 = 4 × 95, the root is twice √95: 2 × 9.746794 ≈ 19.493589.
Frequently asked questions
What is the square root of 380?
The square root of 380 is 2√95 in simplest radical form, which is about 19.4935886896. The negative root, −19.493589, also squares to 380.
Is the square root of 380 rational or irrational?
Irrational. 380 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 √380 be simplified?
Yes. The largest perfect square dividing 380 is 4, so √380 = √4 × √95 = 2√95.
What is √380 rounded to two decimal places?
√380 ≈ 19.49 to two decimal places (19.5 to one, 19.494 to three). Check: 19.49² = 379.8601, close to 380.