Square Root of 258

√258 at a glance

Exact value

√258

Decimal (10 places)

16.0623784042

Rounded

16.1 · 16.06 · 16.062

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.062378

Prime factorization

2 × 3 × 43

Cube root

6.366097

How to simplify √258

The prime factorization of 258 is 2 × 3 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √258 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 258, 2, 3 and 43 appear an odd number of times, so √258 is irrational and 16.0623784042 is a rounded value.

Where √258 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √258 lies between 16 and 17. 258 is 2 above 256 and 31 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.0606 (0.01% low)
• Tangent from 16, i.e. 16 + 2 ÷ 32: 16.0625 (0% high)
• Tangent from 17, i.e. 17 − 31 ÷ 34: 16.0882 (0.16% high)

For √258 the tangent at 16 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 258 is just 2 above 256.

Finding √258 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 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √258 = 16.0623784042 to every decimal shown.

√258 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √258 the pattern is [16; 16, 32] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √258 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 − 258y² = 1. Its smallest solution in positive whole numbers is x = 257, y = 16.

√258 in geometry and everyday measurements

• A square patio or deck of 258 square feet is about 16.06 ft (16 ft 1 in) on each side, so edging all the way around takes 4 × √258 ≈ 64.2 ft.
• 258 is not a sum of two whole-number squares — the prime factor 3 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √258 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 16 box, because 1² + 1² + 16² = 258.

Square roots near √258 and related roots

• The cube root of 258 is about 6.366097.
• Squaring undoes the root: (√258)² = 258, while 258² = 66,564 — the number whose square root is 258.

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