Square Root of 363

√363 at a glance

Exact value

11√3

Decimal (10 places)

19.0525588833

Rounded

19.1 · 19.05 · 19.053

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.052559

Prime factorization

3 × 11²

Cube root

7.133492

How to simplify √363

Look for the largest perfect square that divides 363. Here it is 121 (11²), because 363 = 121 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 363 = 3 × 11². Each pair of equal primes leaves the radical as one factor, so 11 comes out and 3 stays inside.

Check: (11√3)² = 11² × 3 = 121 × 3 = 363. As a decimal, 11√3 = 11 × 1.7320508076 ≈ 19.0525588833.

Where √363 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √363 lies between 19 and 20. 363 is 2 above 361 and 37 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.0513 (0.01% low)
• Tangent from 19, i.e. 19 + 2 ÷ 38: 19.0526 (0% high)
• Tangent from 20, i.e. 20 − 37 ÷ 40: 19.0750 (0.12% high)

For √363 the tangent at 19 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 363 is just 2 above 361.

Finding √363 with the Babylonian method

If a guess is too big, 363 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√363) in one step.

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

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √363 = 19.0525588833 to every decimal shown.

√363 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √363 the pattern is [19; 19, 38] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √363 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 − 363y² = 1. Its smallest solution in positive whole numbers is x = 362, y = 19.

√363 in geometry and everyday measurements

• A square patio or deck of 363 square feet is about 19.05 ft (19 ft 1 in) on each side, so edging all the way around takes 4 × √363 ≈ 76.2 ft.
• 363 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √363 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 19 box, because 1² + 1² + 19² = 363.
• Since √363 = 11√3, a length of √363 is exactly 11 copies of the length √3 laid end to end.

Square roots near √363 and related roots

• The cube root of 363 is about 7.133492.
• Squaring undoes the root: (√363)² = 363, while 363² = 131,769 — the number whose square root is 363.

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