Square Root of 664

√664 at a glance

Exact value

2√166

Decimal (10 places)

25.7681974535

Rounded

25.8 · 25.77 · 25.768

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.768197

Prime factorization

2³ × 83

Cube root

8.724141

How to simplify √664

Look for the largest perfect square that divides 664. Here it is 4 (2²), because 664 = 4 × 166 and 166 has no square factor left:

The prime factorization tells the same story: 664 = 2³ × 83. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 83 stays inside.

Check: (2√166)² = 2² × 166 = 4 × 166 = 664. As a decimal, 2√166 = 2 × 12.8840987267 ≈ 25.7681974535.

Where √664 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √664 lies between 25 and 26. 664 is 39 above 625 and 12 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.7647 (0.01% low)
• Tangent from 25, i.e. 25 + 39 ÷ 50: 25.7800 (0.05% high)
• Tangent from 26, i.e. 26 − 12 ÷ 52: 25.7692 (0% high)

For √664 the tangent at 26 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 664 is just 12 below 676.

Finding √664 with the Babylonian method

Picture a rectangle with an area of 664 and one side x; the other side must be 664 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √664.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √664 = 25.7681974535 to every decimal shown.

√664 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √664 the pattern is [25; 1, 3, 3, 5, 2, 2, 1, 1, 2, 1, 5, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √664 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 − 664y² = 1. Its smallest solution in positive whole numbers is x = 1,700,902,565, y = 66,007,821.

√664 in geometry and everyday measurements

• 664 square feet is 61.7 m². Laid out as a square — a small house footprint or a lot — it is about 25.77 ft (25 ft 9 in) on a side.
• 664 is not a sum of two whole-number squares — the prime factor 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √664 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 18 × 18 box, because 4² + 18² + 18² = 664.
• Since √664 = 2√166, a length of √664 is exactly 2 copies of the length √166 laid end to end.

Square roots near √664 and related roots

• The cube root of 664 is about 8.724141.
• Because 664 = 4 × 166, the root is twice √166: 2 × 12.884099 ≈ 25.768197.

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