Square Root of 492

√492 at a glance

Exact value

2√123

Decimal (10 places)

22.1810730128

Rounded

22.2 · 22.18 · 22.181

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.181073

Prime factorization

2² × 3 × 41

Cube root

7.894447

How to simplify √492

Look for the largest perfect square that divides 492. Here it is 4 (2²), because 492 = 4 × 123 and 123 has no square factor left:

The prime factorization tells the same story: 492 = 2² × 3 × 41. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 41 stays inside.

Check: (2√123)² = 2² × 123 = 4 × 123 = 492. As a decimal, 2√123 = 2 × 11.0905365064 ≈ 22.1810730128.

Where √492 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √492 lies between 22 and 23. 492 is 8 above 484 and 37 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.1778 (0.01% low)
• Tangent from 22, i.e. 22 + 8 ÷ 44: 22.1818 (0% high)
• Tangent from 23, i.e. 23 − 37 ÷ 46: 22.1957 (0.07% high)

For √492 the tangent at 22 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 492 is just 8 above 484.

Finding √492 with the Babylonian method

Picture a rectangle with an area of 492 and one side x; the other side must be 492 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √492.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √492 = 22.1810730128 to every decimal shown.

√492 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √492 the pattern is [22; 5, 1, 1, 10, 1, 1, 5, 44] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √492 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 − 492y² = 1. Its smallest solution in positive whole numbers is x = 29,767, y = 1,342.

√492 in geometry and everyday measurements

• A square garage floor of 492 square feet measures about 22.18 ft (22 ft 2 in) per side, and its corner-to-corner diagonal is √984 ≈ 31.4 ft.
• 492 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √492 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 22 box, because 2² + 2² + 22² = 492.
• Since √492 = 2√123, a length of √492 is exactly 2 copies of the length √123 laid end to end.

Square roots near √492 and related roots

• The cube root of 492 is about 7.894447.
• Because 492 = 4 × 123, the root is twice √123: 2 × 11.090537 ≈ 22.181073.

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