Square Root of 92

√92 at a glance

Exact value

2√23

Decimal (10 places)

9.5916630466

Rounded

9.6 · 9.59 · 9.592

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.591663

Prime factorization

2² × 23

Cube root

4.514357

How to simplify √92

Look for the largest perfect square that divides 92. Here it is 4 (2²), because 92 = 4 × 23 and 23 has no square factor left:

The prime factorization tells the same story: 92 = 2² × 23. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 23 stays inside.

Check: (2√23)² = 2² × 23 = 4 × 23 = 92. As a decimal, 2√23 = 2 × 4.7958315233 ≈ 9.5916630466.

Where √92 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √92 lies between 9 and 10. 92 is 11 above 81 and 8 below 100, so the root is closer to 10.

• Straight line between 81 and 100: 9.5789 (0.13% low)
• Tangent from 9, i.e. 9 + 11 ÷ 18: 9.6111 (0.2% high)
• Tangent from 10, i.e. 10 − 8 ÷ 20: 9.6000 (0.09% high)

For √92 the tangent at 10 wins, missing by only 0.0083. Tangent estimates shine when the number sits close to a perfect square — here 92 is just 8 below 100.

Finding √92 with the Babylonian method

Picture a rectangle with an area of 92 and one side x; the other side must be 92 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √92.

Start from the nearest whole number, 10 (10² = 100):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √92 = 9.5916630466 to every decimal shown.

√92 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √92 the pattern is [9; 1, 1, 2, 4, 2, 1, 1, 18] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √92 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 − 92y² = 1. Its smallest solution in positive whole numbers is x = 1,151, y = 120.

√92 in geometry and everyday measurements

• A square room or garden bed covering 92 square feet measures about 9.59 ft (9 ft 7 in) along each wall.
• 92 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √92 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √92 as its space diagonal.
• Since √92 = 2√23, a length of √92 is exactly 2 copies of the length √23 laid end to end.

Square roots near √92 and related roots

• The cube root of 92 is about 4.514357.
• Four times the radicand doubles the root: √368 = 2 × √92 ≈ 19.183326.

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