Square Root of 662

√662 at a glance

Exact value

√662

Decimal (10 places)

25.7293606605

Rounded

25.7 · 25.73 · 25.729

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.729361

Prime factorization

2 × 331

Cube root

8.715373

How to simplify √662

The prime factorization of 662 is 2 × 331. Every prime appears only once, so there is no pair to bring outside the radical — √662 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 662, 2 and 331 appear an odd number of times, so √662 is irrational and 25.7293606605 is a rounded value.

Where √662 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √662 lies between 25 and 26. 662 is 37 above 625 and 14 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.7255 (0.02% low)
• Tangent from 25, i.e. 25 + 37 ÷ 50: 25.7400 (0.04% high)
• Tangent from 26, i.e. 26 − 14 ÷ 52: 25.7308 (0.01% high)

For √662 the tangent at 26 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 662 is just 14 below 676.

Finding √662 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, 26 (26² = 676):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √662 = 25.7293606605 to every decimal shown.

√662 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √662 the pattern is [25; 1, 2, 1, 2, 3, 1, 1, 2, 6, 1, 24, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √662 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 − 662y² = 1. Its smallest solution in positive whole numbers is x = 1,718,102,501, y = 66,775,950.

√662 in geometry and everyday measurements

• 662 square feet is 61.5 m². Laid out as a square — a small house footprint or a lot — it is about 25.73 ft (25 ft 9 in) on a side.
• 662 is not a sum of two whole-number squares — the prime factor 331 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √662 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 25 box, because 1² + 6² + 25² = 662.

Square roots near √662 and related roots

• The cube root of 662 is about 8.715373.
• Squaring undoes the root: (√662)² = 662, while 662² = 438,244 — the number whose square root is 662.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.