Square Root of 644

√644 at a glance

Exact value

2√161

Decimal (10 places)

25.3771550809

Rounded

25.4 · 25.38 · 25.377

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.377155

Prime factorization

2² × 7 × 23

Cube root

8.635655

How to simplify √644

Look for the largest perfect square that divides 644. Here it is 4 (2²), because 644 = 4 × 161 and 161 has no square factor left:

The prime factorization tells the same story: 644 = 2² × 7 × 23. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 × 23 stays inside.

Check: (2√161)² = 2² × 161 = 4 × 161 = 644. As a decimal, 2√161 = 2 × 12.6885775404 ≈ 25.3771550809.

Where √644 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √644 lies between 25 and 26. 644 is 19 above 625 and 32 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.3725 (0.02% low)
• Tangent from 25, i.e. 25 + 19 ÷ 50: 25.3800 (0.01% high)
• Tangent from 26, i.e. 26 − 32 ÷ 52: 25.3846 (0.03% high)

For √644 the tangent at 25 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 644 is just 19 above 625.

Finding √644 with the Babylonian method

Picture a rectangle with an area of 644 and one side x; the other side must be 644 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √644.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √644 = 25.3771550809 to every decimal shown.

√644 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √644 the pattern is [25; 2, 1, 1, 1, 6, 1, 1, 1, 2, 50] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √644 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 − 644y² = 1. Its smallest solution in positive whole numbers is x = 11,775, y = 464.

√644 in geometry and everyday measurements

• A square garage floor of 644 square feet measures about 25.38 ft (25 ft 5 in) per side, and its corner-to-corner diagonal is √1288 ≈ 35.9 ft.
• 644 is not a sum of two whole-number squares — the prime factor 7 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √644 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 24 box, because 2² + 8² + 24² = 644.
• Since √644 = 2√161, a length of √644 is exactly 2 copies of the length √161 laid end to end.

Square roots near √644 and related roots

• The cube root of 644 is about 8.635655.
• Because 644 = 4 × 161, the root is twice √161: 2 × 12.688578 ≈ 25.377155.

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