Square Root of 681

√681 at a glance

Exact value

√681

Decimal (10 places)

26.0959767014

Rounded

26.1 · 26.10 · 26.096

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.095977

Prime factorization

3 × 227

Cube root

8.797968

How to simplify √681

The prime factorization of 681 is 3 × 227. Every prime appears only once, so there is no pair to bring outside the radical — √681 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 681, 3 and 227 appear an odd number of times, so √681 is irrational and 26.0959767014 is a rounded value.

Where √681 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √681 lies between 26 and 27. 681 is 5 above 676 and 48 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.0943 (0.01% low)
• Tangent from 26, i.e. 26 + 5 ÷ 52: 26.0962 (0% high)
• Tangent from 27, i.e. 27 − 48 ÷ 54: 26.1111 (0.06% high)

For √681 the tangent at 26 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 681 is just 5 above 676.

Finding √681 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 681: following the tangent line down to zero simplifies to averaging x with 681 ÷ x.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √681 = 26.0959767014 to every decimal shown.

√681 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √681 the pattern is [26; 10, 2, 2, 1, 1, 2, 6, 7, 3, 2, 1, 16, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √681 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 − 681y² = 1. Its smallest solution in positive whole numbers is x = 10,743,166,003,415, y = 411,679,015,748 — 14 digits for x, even though 681 is small, which is what makes Pell’s equation famous.

√681 in geometry and everyday measurements

• 681 square feet is 63.3 m². Laid out as a square — a small house footprint or a lot — it is about 26.1 ft (26 ft 1 in) on a side.
• 681 is not a sum of two whole-number squares — the prime factor 3 and 227 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √681 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 26 box, because 1² + 2² + 26² = 681.

Square roots near √681 and related roots

• The cube root of 681 is about 8.797968.
• Squaring undoes the root: (√681)² = 681, while 681² = 463,761 — the number whose square root is 681.

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