Square Root of 660

√660 at a glance

Exact value

2√165

Decimal (10 places)

25.6904651573

Rounded

25.7 · 25.69 · 25.690

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.690465

Prime factorization

2² × 3 × 5 × 11

Cube root

8.706588

How to simplify √660

Look for the largest perfect square that divides 660. Here it is 4 (2²), because 660 = 4 × 165 and 165 has no square factor left:

The prime factorization tells the same story: 660 = 2² × 3 × 5 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 5 × 11 stays inside.

Check: (2√165)² = 2² × 165 = 4 × 165 = 660. As a decimal, 2√165 = 2 × 12.8452325787 ≈ 25.6904651573.

Where √660 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √660 lies between 25 and 26. 660 is 35 above 625 and 16 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.6863 (0.02% low)
• Tangent from 25, i.e. 25 + 35 ÷ 50: 25.7000 (0.04% high)
• Tangent from 26, i.e. 26 − 16 ÷ 52: 25.6923 (0.01% high)

For √660 the tangent at 26 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 660 is just 16 below 676.

Finding √660 with the Babylonian method

Picture a rectangle with an area of 660 and one side x; the other side must be 660 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √660.

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 √660 = 25.6904651573 to every decimal shown.

√660 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √660 the pattern is [25; 1, 2, 4, 2, 1, 50] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √660 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 − 660y² = 1. Its smallest solution in positive whole numbers is x = 1,079, y = 42.

√660 in geometry and everyday measurements

• 660 square feet is 61.3 m². Laid out as a square — a small house footprint or a lot — it is about 25.69 ft (25 ft 8 in) on a side.
• 660 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 √660 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 16 × 20 box, because 2² + 16² + 20² = 660.
• Since √660 = 2√165, a length of √660 is exactly 2 copies of the length √165 laid end to end.

Square roots near √660 and related roots

• The cube root of 660 is about 8.706588.
• Because 660 = 4 × 165, the root is twice √165: 2 × 12.845233 ≈ 25.690465.

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