Square Root of 694

√694 at a glance

Exact value

√694

Decimal (10 places)

26.3438797446

Rounded

26.3 · 26.34 · 26.344

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.343880

Prime factorization

2 × 347

Cube root

8.853599

How to simplify √694

The prime factorization of 694 is 2 × 347. Every prime appears only once, so there is no pair to bring outside the radical — √694 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 694, 2 and 347 appear an odd number of times, so √694 is irrational and 26.3438797446 is a rounded value.

Where √694 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √694 lies between 26 and 27. 694 is 18 above 676 and 35 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.3396 (0.02% low)
• Tangent from 26, i.e. 26 + 18 ÷ 52: 26.3462 (0.01% high)
• Tangent from 27, i.e. 27 − 35 ÷ 54: 26.3519 (0.03% high)

For √694 the tangent at 26 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 694 is just 18 above 676.

Finding √694 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 √694 = 26.3438797446 to every decimal shown.

√694 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √694 the pattern is [26; 2, 1, 9, 1, 6, 1, 1, 1, 1, 1, 2, 1, …] with the block of 46 terms after the semicolon repeating forever (only the first 12 of the 46 are shown). A pattern that never ends is one more proof that √694 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 − 694y² = 1. Its smallest solution in positive whole numbers is x = 38,782,105,445,014,642,382,885, y = 1,472,148,590,903,997,672,114 — 23 digits for x, even though 694 is small, which is what makes Pell’s equation famous.

√694 in geometry and everyday measurements

• 694 square feet is 64.5 m². Laid out as a square — a small house footprint or a lot — it is about 26.34 ft (26 ft 4 in) on a side.
• 694 is not a sum of two whole-number squares — the prime factor 347 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √694 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 26 box, because 3² + 3² + 26² = 694.

Square roots near √694 and related roots

• The cube root of 694 is about 8.853599.
• Squaring undoes the root: (√694)² = 694, while 694² = 481,636 — the number whose square root is 694.

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