Square Root of 374

√374 at a glance

Exact value

√374

Decimal (10 places)

19.3390796058

Rounded

19.3 · 19.34 · 19.339

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.339080

Prime factorization

2 × 11 × 17

Cube root

7.204832

How to simplify √374

The prime factorization of 374 is 2 × 11 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √374 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 374, 2, 11 and 17 appear an odd number of times, so √374 is irrational and 19.3390796058 is a rounded value.

Where √374 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √374 lies between 19 and 20. 374 is 13 above 361 and 26 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.3333 (0.03% low)
• Tangent from 19, i.e. 19 + 13 ÷ 38: 19.3421 (0.02% high)
• Tangent from 20, i.e. 20 − 26 ÷ 40: 19.3500 (0.06% high)

For √374 the tangent at 19 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 374 is just 13 above 361.

Finding √374 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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 √374 = 19.3390796058 to every decimal shown.

√374 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √374 the pattern is [19; 2, 1, 18, 1, 2, 38] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √374 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 − 374y² = 1. Its smallest solution in positive whole numbers is x = 3,365, y = 174.

√374 in geometry and everyday measurements

• A square patio or deck of 374 square feet is about 19.34 ft (19 ft 4 in) on each side, so edging all the way around takes 4 × √374 ≈ 77.4 ft.
• 374 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √374 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 18 box, because 1² + 7² + 18² = 374.

Square roots near √374 and related roots

• The cube root of 374 is about 7.204832.
• Squaring undoes the root: (√374)² = 374, while 374² = 139,876 — the number whose square root is 374.

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