Square Root of 377

√377 at a glance

Exact value

√377

Decimal (10 places)

19.4164878389

Rounded

19.4 · 19.42 · 19.416

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.416488

Prime factorization

13 × 29

Cube root

7.224045

How to simplify √377

The prime factorization of 377 is 13 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √377 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 377, 13 and 29 appear an odd number of times, so √377 is irrational and 19.4164878389 is a rounded value.

Where √377 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √377 lies between 19 and 20. 377 is 16 above 361 and 23 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.4103 (0.03% low)
• Tangent from 19, i.e. 19 + 16 ÷ 38: 19.4211 (0.02% high)
• Tangent from 20, i.e. 20 − 23 ÷ 40: 19.4250 (0.04% high)

For √377 the tangent at 19 wins, missing by only 0.0046. Tangent estimates shine when the number sits close to a perfect square — here 377 is just 16 above 361.

Finding √377 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 377: following the tangent line down to zero simplifies to averaging x with 377 ÷ x.

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 √377 = 19.4164878389 to every decimal shown.

√377 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √377 the pattern is [19; 2, 2, 2, 38] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √377 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 − 377y² = 1. Its smallest solution in positive whole numbers is x = 233, y = 12.

√377 in geometry and everyday measurements

• A square patio or deck of 377 square feet is about 19.42 ft (19 ft 5 in) on each side, so edging all the way around takes 4 × √377 ≈ 77.7 ft.
• 377 = 4² + 19² = 11² + 16², so by the Pythagorean theorem √377 is the diagonal of rectangles measuring 4 × 19 and 11 × 16 — and the distance between the points (0, 0) and (4, 19) on a grid.

Square roots near √377 and related roots

• The cube root of 377 is about 7.224045.
• Squaring undoes the root: (√377)² = 377, while 377² = 142,129 — the number whose square root is 377.

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