Square Root of 540

√540 at a glance

Exact value

6√15

Decimal (10 places)

23.2379000772

Rounded

23.2 · 23.24 · 23.238

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.237900

Prime factorization

2² × 3³ × 5

Cube root

8.143253

How to simplify √540

Look for the largest perfect square that divides 540. Here it is 36 (6²), because 540 = 36 × 15 and 15 has no square factor left:

The prime factorization tells the same story: 540 = 2² × 3³ × 5. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 3 × 5 stays inside.

540 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √540 = 2√135, and √135 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√15)² = 6² × 15 = 36 × 15 = 540. As a decimal, 6√15 = 6 × 3.8729833462 ≈ 23.2379000772.

Where √540 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √540 lies between 23 and 24. 540 is 11 above 529 and 36 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.2340 (0.02% low)
• Tangent from 23, i.e. 23 + 11 ÷ 46: 23.2391 (0.01% high)
• Tangent from 24, i.e. 24 − 36 ÷ 48: 23.2500 (0.05% high)

For √540 the tangent at 23 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 540 is just 11 above 529.

Finding √540 with the Babylonian method

Picture a rectangle with an area of 540 and one side x; the other side must be 540 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √540.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √540 = 23.2379000772 to every decimal shown.

√540 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √540 the pattern is [23; 4, 4, 1, 10, 1, 4, 4, 46] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √540 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 − 540y² = 1. Its smallest solution in positive whole numbers is x = 119,071, y = 5,124.

√540 in geometry and everyday measurements

• A square garage floor of 540 square feet measures about 23.24 ft (23 ft 3 in) per side, and its corner-to-corner diagonal is √1080 ≈ 32.9 ft.
• 540 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 √540 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 √540 as its space diagonal.
• Since √540 = 6√15, a length of √540 is exactly 6 copies of the length √15 laid end to end.

Square roots near √540 and related roots

• The cube root of 540 is about 8.143253.
• Because 540 = 4 × 135, the root is twice √135: 2 × 11.61895 ≈ 23.2379.

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