Square Root of 262

√262 at a glance

Exact value

√262

Decimal (10 places)

16.1864140562

Rounded

16.2 · 16.19 · 16.186

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.186414

Prime factorization

2 × 131

Cube root

6.398828

How to simplify √262

The prime factorization of 262 is 2 × 131. Every prime appears only once, so there is no pair to bring outside the radical — √262 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 262, 2 and 131 appear an odd number of times, so √262 is irrational and 16.1864140562 is a rounded value.

Where √262 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √262 lies between 16 and 17. 262 is 6 above 256 and 27 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.1818 (0.03% low)
• Tangent from 16, i.e. 16 + 6 ÷ 32: 16.1875 (0.01% high)
• Tangent from 17, i.e. 17 − 27 ÷ 34: 16.2059 (0.12% high)

For √262 the tangent at 16 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 262 is just 6 above 256.

Finding √262 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, 16 (16² = 256):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √262 = 16.1864140562 to every decimal shown.

√262 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √262 the pattern is [16; 5, 2, 1, 2, 1, 10, 16, 10, 1, 2, 1, 2, …] 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 √262 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 − 262y² = 1. Its smallest solution in positive whole numbers is x = 104,980,517, y = 6,485,718.

√262 in geometry and everyday measurements

• A square patio or deck of 262 square feet is about 16.19 ft (16 ft 2 in) on each side, so edging all the way around takes 4 × √262 ≈ 64.7 ft.
• 262 is not a sum of two whole-number squares — the prime factor 131 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √262 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 15 box, because 1² + 6² + 15² = 262.

Square roots near √262 and related roots

• The cube root of 262 is about 6.398828.
• Squaring undoes the root: (√262)² = 262, while 262² = 68,644 — the number whose square root is 262.

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