Square Root of 685

√685 at a glance

Exact value

√685

Decimal (10 places)

26.1725046566

Rounded

26.2 · 26.17 · 26.173

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.172505

Prime factorization

5 × 137

Cube root

8.815160

How to simplify √685

The prime factorization of 685 is 5 × 137. Every prime appears only once, so there is no pair to bring outside the radical — √685 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 685, 5 and 137 appear an odd number of times, so √685 is irrational and 26.1725046566 is a rounded value.

Where √685 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √685 lies between 26 and 27. 685 is 9 above 676 and 44 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.1698 (0.01% low)
• Tangent from 26, i.e. 26 + 9 ÷ 52: 26.1731 (0% high)
• Tangent from 27, i.e. 27 − 44 ÷ 54: 26.1852 (0.05% high)

For √685 the tangent at 26 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 685 is just 9 above 676.

Finding √685 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 685: following the tangent line down to zero simplifies to averaging x with 685 ÷ x.

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 √685 = 26.1725046566 to every decimal shown.

√685 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √685 the pattern is [26; 5, 1, 3, 1, 12, 3, 2, 2, 3, 12, 1, 3, …] with the block of 15 terms after the semicolon repeating forever (only the first 12 of the 15 are shown). A pattern that never ends is one more proof that √685 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 − 685y² = 1. Its smallest solution in positive whole numbers is x = 95,592,800,063,517,769, y = 3,652,413,145,693,884 — 17 digits for x, even though 685 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 218,623,878² − 685 × 8,353,189² = −1.

√685 in geometry and everyday measurements

• 685 square feet is 63.6 m². Laid out as a square — a small house footprint or a lot — it is about 26.17 ft (26 ft 2 in) on a side.
• 685 = 3² + 26² = 18² + 19², so by the Pythagorean theorem √685 is the diagonal of rectangles measuring 3 × 26 and 18 × 19 — and the distance between the points (0, 0) and (3, 26) on a grid.

Square roots near √685 and related roots

• The cube root of 685 is about 8.815160.
• Squaring undoes the root: (√685)² = 685, while 685² = 469,225 — the number whose square root is 685.

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