Square Root of 376

√376 at a glance

Exact value

2√94

Decimal (10 places)

19.3907194297

Rounded

19.4 · 19.39 · 19.391

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.390719

Prime factorization

2³ × 47

Cube root

7.217652

How to simplify √376

Look for the largest perfect square that divides 376. Here it is 4 (2²), because 376 = 4 × 94 and 94 has no square factor left:

The prime factorization tells the same story: 376 = 2³ × 47. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 47 stays inside.

Check: (2√94)² = 2² × 94 = 4 × 94 = 376. As a decimal, 2√94 = 2 × 9.6953597148 ≈ 19.3907194297.

Where √376 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √376 lies between 19 and 20. 376 is 15 above 361 and 24 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.3846 (0.03% low)
• Tangent from 19, i.e. 19 + 15 ÷ 38: 19.3947 (0.02% high)
• Tangent from 20, i.e. 20 − 24 ÷ 40: 19.4000 (0.05% high)

For √376 the tangent at 19 wins, missing by only 0.004. Tangent estimates shine when the number sits close to a perfect square — here 376 is just 15 above 361.

Finding √376 with the Babylonian method

Picture a rectangle with an area of 376 and one side x; the other side must be 376 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √376.

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 √376 = 19.3907194297 to every decimal shown.

√376 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √376 the pattern is [19; 2, 1, 1, 3, 1, 2, 2, 4, 2, 2, 1, 3, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √376 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 − 376y² = 1. Its smallest solution in positive whole numbers is x = 2,143,295, y = 110,532.

√376 in geometry and everyday measurements

• A square patio or deck of 376 square feet is about 19.39 ft (19 ft 5 in) on each side, so edging all the way around takes 4 × √376 ≈ 77.6 ft.
• 376 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √376 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 6 × 18 box, because 4² + 6² + 18² = 376.
• Since √376 = 2√94, a length of √376 is exactly 2 copies of the length √94 laid end to end.

Square roots near √376 and related roots

• The cube root of 376 is about 7.217652.
• Because 376 = 4 × 94, the root is twice √94: 2 × 9.69536 ≈ 19.390719.

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