Square Root of 85

√85 at a glance

Exact value

√85

Decimal (10 places)

9.2195444573

Rounded

9.2 · 9.22 · 9.220

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.219544

Prime factorization

5 × 17

Cube root

4.396830

How to simplify √85

The prime factorization of 85 is 5 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √85 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 85, 5 and 17 appear an odd number of times, so √85 is irrational and 9.2195444573 is a rounded value.

Where √85 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √85 lies between 9 and 10. 85 is 4 above 81 and 15 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.2105 (0.1% low)
• Tangent from 9, i.e. 9 + 4 ÷ 18: 9.2222 (0.03% high)
• Tangent from 10, i.e. 10 − 15 ÷ 20: 9.2500 (0.33% high)

For √85 the tangent at 9 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 85 is just 4 above 81.

Finding √85 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 85: following the tangent line down to zero simplifies to averaging x with 85 ÷ x.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √85 = 9.2195444573 to every decimal shown.

√85 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √85 the pattern is [9; 4, 1, 1, 4, 18] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √85 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 − 85y² = 1. Its smallest solution in positive whole numbers is x = 285,769, y = 30,996. Because the period is odd, the equation with −1 on the right also has a solution: 378² − 85 × 41² = −1.

√85 in geometry and everyday measurements

• A square room or garden bed covering 85 square feet measures about 9.22 ft (9 ft 3 in) along each wall.
• 85 = 2² + 9² = 6² + 7², so by the Pythagorean theorem √85 is the diagonal of rectangles measuring 2 × 9 and 6 × 7 — and the distance between the points (0, 0) and (2, 9) on a grid.

Square roots near √85 and related roots

• The cube root of 85 is about 4.396830.
• Four times the radicand doubles the root: √340 = 2 × √85 ≈ 18.439089.

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