Square Root of 369

√369 at a glance

Exact value

3√41

Decimal (10 places)

19.2093727123

Rounded

19.2 · 19.21 · 19.209

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.209373

Prime factorization

3² × 41

Cube root

7.172581

How to simplify √369

Look for the largest perfect square that divides 369. Here it is 9 (3²), because 369 = 9 × 41 and 41 has no square factor left:

The prime factorization tells the same story: 369 = 3² × 41. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 41 stays inside.

Check: (3√41)² = 3² × 41 = 9 × 41 = 369. As a decimal, 3√41 = 3 × 6.4031242374 ≈ 19.2093727123.

Where √369 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √369 lies between 19 and 20. 369 is 8 above 361 and 31 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.2051 (0.02% low)
• Tangent from 19, i.e. 19 + 8 ÷ 38: 19.2105 (0.01% high)
• Tangent from 20, i.e. 20 − 31 ÷ 40: 19.2250 (0.08% high)

For √369 the tangent at 19 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 369 is just 8 above 361.

Finding √369 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 369: following the tangent line down to zero simplifies to averaging x with 369 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √369 = 19.2093727123 to every decimal shown.

√369 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √369 the pattern is [19; 4, 1, 3, 2, 7, 4, 7, 2, 3, 1, 4, 38] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √369 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 − 369y² = 1. Its smallest solution in positive whole numbers is x = 8,396,801, y = 437,120.

√369 in geometry and everyday measurements

• A square patio or deck of 369 square feet is about 19.21 ft (19 ft 3 in) on each side, so edging all the way around takes 4 × √369 ≈ 76.8 ft.
• 369 = 12² + 15², so by the Pythagorean theorem √369 is the diagonal of a 12 × 15 rectangle — and the distance between the points (0, 0) and (12, 15) on a grid.
• Since √369 = 3√41, a length of √369 is exactly 3 copies of the length √41 laid end to end.

Square roots near √369 and related roots

• The cube root of 369 is about 7.172581.
• Squaring undoes the root: (√369)² = 369, while 369² = 136,161 — the number whose square root is 369.

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