Square Root of 372

√372 at a glance

Exact value

2√93

Decimal (10 places)

19.2873015220

Rounded

19.3 · 19.29 · 19.287

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.287302

Prime factorization

2² × 3 × 31

Cube root

7.191966

How to simplify √372

Look for the largest perfect square that divides 372. Here it is 4 (2²), because 372 = 4 × 93 and 93 has no square factor left:

The prime factorization tells the same story: 372 = 2² × 3 × 31. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 31 stays inside.

Check: (2√93)² = 2² × 93 = 4 × 93 = 372. As a decimal, 2√93 = 2 × 9.643650761 ≈ 19.2873015220.

Where √372 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √372 lies between 19 and 20. 372 is 11 above 361 and 28 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.2821 (0.03% low)
• Tangent from 19, i.e. 19 + 11 ÷ 38: 19.2895 (0.01% high)
• Tangent from 20, i.e. 20 − 28 ÷ 40: 19.3000 (0.07% high)

For √372 the tangent at 19 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 372 is just 11 above 361.

Finding √372 with the Babylonian method

Picture a rectangle with an area of 372 and one side x; the other side must be 372 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √372.

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 √372 = 19.2873015220 to every decimal shown.

√372 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √372 the pattern is [19; 3, 2, 12, 2, 3, 38] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √372 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 − 372y² = 1. Its smallest solution in positive whole numbers is x = 12,151, y = 630.

√372 in geometry and everyday measurements

• A square patio or deck of 372 square feet is about 19.29 ft (19 ft 3 in) on each side, so edging all the way around takes 4 × √372 ≈ 77.1 ft.
• 372 is not a sum of two whole-number squares — the prime factor 3 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √372 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 10 × 16 box, because 4² + 10² + 16² = 372.
• Since √372 = 2√93, a length of √372 is exactly 2 copies of the length √93 laid end to end.

Square roots near √372 and related roots

• The cube root of 372 is about 7.191966.
• Because 372 = 4 × 93, the root is twice √93: 2 × 9.643651 ≈ 19.287302.

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