Square Root of 682

√682 at a glance

Exact value

√682

Decimal (10 places)

26.1151297144

Rounded

26.1 · 26.12 · 26.115

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.115130

Prime factorization

2 × 11 × 31

Cube root

8.802272

How to simplify √682

The prime factorization of 682 is 2 × 11 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √682 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 682, 2, 11 and 31 appear an odd number of times, so √682 is irrational and 26.1151297144 is a rounded value.

Where √682 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √682 lies between 26 and 27. 682 is 6 above 676 and 47 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.1132 (0.01% low)
• Tangent from 26, i.e. 26 + 6 ÷ 52: 26.1154 (0% high)
• Tangent from 27, i.e. 27 − 47 ÷ 54: 26.1296 (0.06% high)

For √682 the tangent at 26 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 682 is just 6 above 676.

Finding √682 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 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √682 = 26.1151297144 to every decimal shown.

√682 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √682 the pattern is [26; 8, 1, 2, 5, 2, 5, 2, 1, 8, 52] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √682 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 − 682y² = 1. Its smallest solution in positive whole numbers is x = 1,197,901, y = 45,870.

√682 in geometry and everyday measurements

• 682 square feet is 63.4 m². Laid out as a square — a small house footprint or a lot — it is about 26.12 ft (26 ft 1 in) on a side.
• 682 is not a sum of two whole-number squares — the prime factor 11 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √682 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 12 × 23 box, because 3² + 12² + 23² = 682.

Square roots near √682 and related roots

• The cube root of 682 is about 8.802272.
• Squaring undoes the root: (√682)² = 682, while 682² = 465,124 — the number whose square root is 682.

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