Square Root of 356

√356 at a glance

Exact value

2√89

Decimal (10 places)

18.8679622641

Rounded

18.9 · 18.87 · 18.868

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.867962

Prime factorization

2² × 89

Cube root

7.087341

How to simplify √356

Look for the largest perfect square that divides 356. Here it is 4 (2²), because 356 = 4 × 89 and 89 has no square factor left:

The prime factorization tells the same story: 356 = 2² × 89. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 89 stays inside.

Check: (2√89)² = 2² × 89 = 4 × 89 = 356. As a decimal, 2√89 = 2 × 9.4339811321 ≈ 18.8679622641.

Where √356 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √356 lies between 18 and 19. 356 is 32 above 324 and 5 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.8649 (0.02% low)
• Tangent from 18, i.e. 18 + 32 ÷ 36: 18.8889 (0.11% high)
• Tangent from 19, i.e. 19 − 5 ÷ 38: 18.8684 (0% high)

For √356 the tangent at 19 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 356 is just 5 below 361.

Finding √356 with the Babylonian method

Picture a rectangle with an area of 356 and one side x; the other side must be 356 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √356.

Start from the nearest whole number, 19 (19² = 361):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √356 = 18.8679622641 to every decimal shown.

√356 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √356 the pattern is [18; 1, 6, 1, 1, 2, 1, 8, 1, 2, 1, 1, 6, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √356 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 − 356y² = 1. Its smallest solution in positive whole numbers is x = 500,001, y = 26,500.

√356 in geometry and everyday measurements

• A square patio or deck of 356 square feet is about 18.87 ft (18 ft 10 in) on each side, so edging all the way around takes 4 × √356 ≈ 75.5 ft.
• 356 = 10² + 16², so by the Pythagorean theorem √356 is the diagonal of a 10 × 16 rectangle — and the distance between the points (0, 0) and (10, 16) on a grid.
• Since √356 = 2√89, a length of √356 is exactly 2 copies of the length √89 laid end to end.

Square roots near √356 and related roots

• The cube root of 356 is about 7.087341.
• Because 356 = 4 × 89, the root is twice √89: 2 × 9.433981 ≈ 18.867962.

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