Square Root of 396

√396 at a glance

Exact value

6√11

Decimal (10 places)

19.8997487421

Rounded

19.9 · 19.90 · 19.900

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.899749

Prime factorization

2² × 3² × 11

Cube root

7.343420

How to simplify √396

Look for the largest perfect square that divides 396. Here it is 36 (6²), because 396 = 36 × 11 and 11 has no square factor left:

The prime factorization tells the same story: 396 = 2² × 3² × 11. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 11 stays inside.

396 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √396 = 2√99, and √99 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√11)² = 6² × 11 = 36 × 11 = 396. As a decimal, 6√11 = 6 × 3.3166247904 ≈ 19.8997487421.

Where √396 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √396 lies between 19 and 20. 396 is 35 above 361 and 4 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.8974 (0.01% low)
• Tangent from 19, i.e. 19 + 35 ÷ 38: 19.9211 (0.11% high)
• Tangent from 20, i.e. 20 − 4 ÷ 40: 19.9000 (0% high)

For √396 the tangent at 20 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 396 is just 4 below 400.

Finding √396 with the Babylonian method

Picture a rectangle with an area of 396 and one side x; the other side must be 396 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √396.

Start from the nearest whole number, 20 (20² = 400):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √396 = 19.8997487421 to every decimal shown.

√396 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √396 the pattern is [19; 1, 8, 1, 38] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √396 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 − 396y² = 1. Its smallest solution in positive whole numbers is x = 199, y = 10.

√396 in geometry and everyday measurements

• A square patio or deck of 396 square feet is about 19.9 ft (19 ft 11 in) on each side, so edging all the way around takes 4 × √396 ≈ 79.6 ft.
• 396 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 √396 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 14 × 14 box, because 2² + 14² + 14² = 396.
• Since √396 = 6√11, a length of √396 is exactly 6 copies of the length √11 laid end to end.

Square roots near √396 and related roots

• The cube root of 396 is about 7.343420.
• Because 396 = 4 × 99, the root is twice √99: 2 × 9.949874 ≈ 19.899749.

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