Square Root of 364

√364 at a glance

Exact value

2√91

Decimal (10 places)

19.0787840283

Rounded

19.1 · 19.08 · 19.079

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.078784

Prime factorization

2² × 7 × 13

Cube root

7.140037

How to simplify √364

Look for the largest perfect square that divides 364. Here it is 4 (2²), because 364 = 4 × 91 and 91 has no square factor left:

The prime factorization tells the same story: 364 = 2² × 7 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 × 13 stays inside.

Check: (2√91)² = 2² × 91 = 4 × 91 = 364. As a decimal, 2√91 = 2 × 9.5393920142 ≈ 19.0787840283.

Where √364 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √364 lies between 19 and 20. 364 is 3 above 361 and 36 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.0769 (0.01% low)
• Tangent from 19, i.e. 19 + 3 ÷ 38: 19.0789 (0% high)
• Tangent from 20, i.e. 20 − 36 ÷ 40: 19.1000 (0.11% high)

For √364 the tangent at 19 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 364 is just 3 above 361.

Finding √364 with the Babylonian method

Picture a rectangle with an area of 364 and one side x; the other side must be 364 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √364.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √364 = 19.0787840283 to every decimal shown.

√364 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √364 the pattern is [19; 12, 1, 2, 3, 1, 8, 1, 3, 2, 1, 12, 38] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √364 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 − 364y² = 1. Its smallest solution in positive whole numbers is x = 4,954,951, y = 259,710.

√364 in geometry and everyday measurements

• A square patio or deck of 364 square feet is about 19.08 ft (19 ft 1 in) on each side, so edging all the way around takes 4 × √364 ≈ 76.3 ft.
• 364 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √364 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 18 box, because 2² + 6² + 18² = 364.
• Since √364 = 2√91, a length of √364 is exactly 2 copies of the length √91 laid end to end.

Square roots near √364 and related roots

• The cube root of 364 is about 7.140037.
• Because 364 = 4 × 91, the root is twice √91: 2 × 9.539392 ≈ 19.078784.

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