Square Root of 54

√54 at a glance

Exact value

3√6

Decimal (10 places)

7.3484692283

Rounded

7.3 · 7.35 · 7.348

Perfect square?

No — between 7² and 8²

Rational?

Irrational

Both square roots

±7.348469

Prime factorization

2 × 3³

Cube root

3.779763

How to simplify √54

Look for the largest perfect square that divides 54. Here it is 9 (3²), because 54 = 9 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 54 = 2 × 3³. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 3 stays inside.

Check: (3√6)² = 3² × 6 = 9 × 6 = 54. As a decimal, 3√6 = 3 × 2.4494897428 ≈ 7.3484692283.

Where √54 sits between perfect squares

49 = 7² and 64 = 8² are the nearest perfect squares, so √54 lies between 7 and 8. 54 is 5 above 49 and 10 below 64, so the root is closer to 7.

• Straight line between 49 and 64: 7.3333 (0.21% low)
• Tangent from 7, i.e. 7 + 5 ÷ 14: 7.3571 (0.12% high)
• Tangent from 8, i.e. 8 − 10 ÷ 16: 7.3750 (0.36% high)

For √54 the tangent at 7 wins, missing by only 0.0087. Tangent estimates shine when the number sits close to a perfect square — here 54 is just 5 above 49.

Finding √54 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, 7 (7² = 49):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √54 = 7.3484692283 to every decimal shown.

√54 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √54 the pattern is [7; 2, 1, 6, 1, 2, 14] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √54 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 − 54y² = 1. Its smallest solution in positive whole numbers is x = 485, y = 66.

√54 in geometry and everyday measurements

• A square room or garden bed covering 54 square feet measures about 7.35 ft (7 ft 4 in) along each wall.
• 54 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 √54 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 7 box, because 1² + 2² + 7² = 54.
• Since √54 = 3√6, a length of √54 is exactly 3 copies of the length √6 laid end to end.

Square roots near √54 and related roots

• The cube root of 54 is about 3.779763.
• Four times the radicand doubles the root: √216 = 2 × √54 ≈ 14.696938.

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