Square Root of 379

√379 at a glance

Exact value

√379

Decimal (10 places)

19.4679223339

Rounded

19.5 · 19.47 · 19.468

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.467922

Prime factorization

379

Cube root

7.236797

How to simplify √379

379 is a prime number, so its only factors are 1 and 379. There is no perfect-square factor to pull out, which means √379 is already in its simplest radical form.

The square root of any prime is irrational. If √379 were a fraction a/b in lowest terms, then a² = 379b², so 379 would divide a — and then 379 would divide b too, contradicting “lowest terms.” That is why the decimal 19.4679223339 is only a rounded value.

Where √379 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √379 lies between 19 and 20. 379 is 18 above 361 and 21 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.4615 (0.03% low)
• Tangent from 19, i.e. 19 + 18 ÷ 38: 19.4737 (0.03% high)
• Tangent from 20, i.e. 20 − 21 ÷ 40: 19.4750 (0.04% high)

For √379 the tangent at 19 wins, missing by only 0.0058. Tangent estimates shine when the number sits close to a perfect square — here 379 is just 18 above 361.

Finding √379 with the Babylonian method

If a guess is too big, 379 ÷ 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√379) in one step.

Start from the nearest whole number, 19 (19² = 361):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √379 = 19.4679223339 to every decimal shown.

√379 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √379 the pattern is [19; 2, 7, 3, 2, 2, 6, 12, 1, 4, 1, 1, 1, …] with the block of 30 terms after the semicolon repeating forever (only the first 12 of the 30 are shown). A pattern that never ends is one more proof that √379 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 379y² = 1. Its smallest solution in positive whole numbers is x = 12,941,197,220,540,690, y = 664,744,650,125,541 — 17 digits for x, even though 379 is small, which is what makes Pell’s equation famous.

√379 in geometry and everyday measurements

• A square patio or deck of 379 square feet is about 19.47 ft (19 ft 6 in) on each side, so edging all the way around takes 4 × √379 ≈ 77.9 ft.
• 379 is not a sum of two whole-number squares — 379 is itself a prime that is one less than a multiple of 4, which rules that out — so √379 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 19 box, because 3² + 3² + 19² = 379.

Square roots near √379 and related roots

• The cube root of 379 is about 7.236797.
• Squaring undoes the root: (√379)² = 379, while 379² = 143,641 — the number whose square root is 379.

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