Square Root of 362

√362 at a glance

Exact value

√362

Decimal (10 places)

19.0262975904

Rounded

19.0 · 19.03 · 19.026

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.026298

Prime factorization

2 × 181

Cube root

7.126936

How to simplify √362

The prime factorization of 362 is 2 × 181. Every prime appears only once, so there is no pair to bring outside the radical — √362 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 362, 2 and 181 appear an odd number of times, so √362 is irrational and 19.0262975904 is a rounded value.

Where √362 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √362 lies between 19 and 20. 362 is 1 above 361 and 38 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.0256 (0% low)
• Tangent from 19, i.e. 19 + 1 ÷ 38: 19.0263 (0% high)
• Tangent from 20, i.e. 20 − 38 ÷ 40: 19.0500 (0.12% high)

For √362 the tangent at 19 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 362 is just 1 above 361.

Finding √362 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

Because the starting guess was already close, two steps are enough to match √362 = 19.0262975904 to every decimal shown.

√362 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √362 the pattern is [19; 38] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 362 is one more than a perfect square (19² + 1). A pattern that never ends is one more proof that √362 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 − 362y² = 1. Its smallest solution in positive whole numbers is x = 723, y = 38. Because the period is odd, the equation with −1 on the right also has a solution: 19² − 362 × 1² = −1.

√362 in geometry and everyday measurements

• A square patio or deck of 362 square feet is about 19.03 ft (19 ft) on each side, so edging all the way around takes 4 × √362 ≈ 76.1 ft.
• 362 = 1² + 19², so by the Pythagorean theorem √362 is the diagonal of a 1 × 19 rectangle — and the distance between the points (0, 0) and (1, 19) on a grid.

Square roots near √362 and related roots

• The cube root of 362 is about 7.126936.
• Squaring undoes the root: (√362)² = 362, while 362² = 131,044 — the number whose square root is 362.

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