Square Root of 332

√332 at a glance

Exact value

2√83

Decimal (10 places)

18.2208671583

Rounded

18.2 · 18.22 · 18.221

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.220867

Prime factorization

2² × 83

Cube root

6.924356

How to simplify √332

Look for the largest perfect square that divides 332. Here it is 4 (2²), because 332 = 4 × 83 and 83 has no square factor left:

The prime factorization tells the same story: 332 = 2² × 83. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 83 stays inside.

Check: (2√83)² = 2² × 83 = 4 × 83 = 332. As a decimal, 2√83 = 2 × 9.1104335791 ≈ 18.2208671583.

Where √332 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √332 lies between 18 and 19. 332 is 8 above 324 and 29 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.2162 (0.03% low)
• Tangent from 18, i.e. 18 + 8 ÷ 36: 18.2222 (0.01% high)
• Tangent from 19, i.e. 19 − 29 ÷ 38: 18.2368 (0.09% high)

For √332 the tangent at 18 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 332 is just 8 above 324.

Finding √332 with the Babylonian method

Picture a rectangle with an area of 332 and one side x; the other side must be 332 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √332.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √332 = 18.2208671583 to every decimal shown.

√332 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √332 the pattern is [18; 4, 1, 1, 8, 1, 1, 4, 36] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √332 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 − 332y² = 1. Its smallest solution in positive whole numbers is x = 13,447, y = 738.

√332 in geometry and everyday measurements

• A square patio or deck of 332 square feet is about 18.22 ft (18 ft 3 in) on each side, so edging all the way around takes 4 × √332 ≈ 72.9 ft.
• 332 is not a sum of two whole-number squares — the prime factor 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √332 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 18 box, because 2² + 2² + 18² = 332.
• Since √332 = 2√83, a length of √332 is exactly 2 copies of the length √83 laid end to end.

Square roots near √332 and related roots

• The cube root of 332 is about 6.924356.
• Because 332 = 4 × 83, the root is twice √83: 2 × 9.110434 ≈ 18.220867.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.