Square Root of 264

√264 at a glance

Exact value

2√66

Decimal (10 places)

16.2480768093

Rounded

16.2 · 16.25 · 16.248

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.248077

Prime factorization

2³ × 3 × 11

Cube root

6.415069

How to simplify √264

Look for the largest perfect square that divides 264. Here it is 4 (2²), because 264 = 4 × 66 and 66 has no square factor left:

The prime factorization tells the same story: 264 = 2³ × 3 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 11 stays inside.

Check: (2√66)² = 2² × 66 = 4 × 66 = 264. As a decimal, 2√66 = 2 × 8.1240384046 ≈ 16.2480768093.

Where √264 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √264 lies between 16 and 17. 264 is 8 above 256 and 25 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.2424 (0.03% low)
• Tangent from 16, i.e. 16 + 8 ÷ 32: 16.2500 (0.01% high)
• Tangent from 17, i.e. 17 − 25 ÷ 34: 16.2647 (0.1% high)

For √264 the tangent at 16 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 264 is just 8 above 256.

Finding √264 with the Babylonian method

Picture a rectangle with an area of 264 and one side x; the other side must be 264 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √264.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √264 = 16.2480768093 to every decimal shown.

√264 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √264 the pattern is [16; 4, 32] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √264 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 − 264y² = 1. Its smallest solution in positive whole numbers is x = 65, y = 4.

√264 in geometry and everyday measurements

• A square patio or deck of 264 square feet is about 16.25 ft (16 ft 3 in) on each side, so edging all the way around takes 4 × √264 ≈ 65 ft.
• 264 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √264 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 16 box, because 2² + 2² + 16² = 264.
• Since √264 = 2√66, a length of √264 is exactly 2 copies of the length √66 laid end to end.

Square roots near √264 and related roots

• The cube root of 264 is about 6.415069.
• Because 264 = 4 × 66, the root is twice √66: 2 × 8.124038 ≈ 16.248077.

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