Square Root of 266

√266 at a glance

Exact value

√266

Decimal (10 places)

16.3095064303

Rounded

16.3 · 16.31 · 16.310

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.309506

Prime factorization

2 × 7 × 19

Cube root

6.431228

How to simplify √266

The prime factorization of 266 is 2 × 7 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √266 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 266, 2, 7 and 19 appear an odd number of times, so √266 is irrational and 16.3095064303 is a rounded value.

Where √266 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √266 lies between 16 and 17. 266 is 10 above 256 and 23 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.3030 (0.04% low)
• Tangent from 16, i.e. 16 + 10 ÷ 32: 16.3125 (0.02% high)
• Tangent from 17, i.e. 17 − 23 ÷ 34: 16.3235 (0.09% high)

For √266 the tangent at 16 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 266 is just 10 above 256.

Finding √266 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, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √266 = 16.3095064303 to every decimal shown.

√266 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √266 the pattern is [16; 3, 4, 3, 32] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √266 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 − 266y² = 1. Its smallest solution in positive whole numbers is x = 685, y = 42.

√266 in geometry and everyday measurements

• A square patio or deck of 266 square feet is about 16.31 ft (16 ft 4 in) on each side, so edging all the way around takes 4 × √266 ≈ 65.2 ft.
• 266 is not a sum of two whole-number squares — the prime factor 7 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √266 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 16 box, because 1² + 3² + 16² = 266.

Square roots near √266 and related roots

• The cube root of 266 is about 6.431228.
• Squaring undoes the root: (√266)² = 266, while 266² = 70,756 — the number whose square root is 266.

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