Square Root of 86

√86 at a glance

Exact value

√86

Decimal (10 places)

9.2736184955

Rounded

9.3 · 9.27 · 9.274

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.273618

Prime factorization

2 × 43

Cube root

4.414005

How to simplify √86

The prime factorization of 86 is 2 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √86 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 86, 2 and 43 appear an odd number of times, so √86 is irrational and 9.2736184955 is a rounded value.

Where √86 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √86 lies between 9 and 10. 86 is 5 above 81 and 14 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.2632 (0.11% low)
• Tangent from 9, i.e. 9 + 5 ÷ 18: 9.2778 (0.04% high)
• Tangent from 10, i.e. 10 − 14 ÷ 20: 9.3000 (0.28% high)

For √86 the tangent at 9 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 86 is just 5 above 81.

Finding √86 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, 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 √86 = 9.2736184955 to every decimal shown.

√86 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √86 the pattern is [9; 3, 1, 1, 1, 8, 1, 1, 1, 3, 18] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √86 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 − 86y² = 1. Its smallest solution in positive whole numbers is x = 10,405, y = 1,122.

√86 in geometry and everyday measurements

• A square room or garden bed covering 86 square feet measures about 9.27 ft (9 ft 3 in) along each wall.
• 86 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √86 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 9 box, because 1² + 2² + 9² = 86.

Square roots near √86 and related roots

• The cube root of 86 is about 4.414005.
• Four times the radicand doubles the root: √344 = 2 × √86 ≈ 18.547237.

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