Square Root of 636

√636 at a glance

Exact value

2√159

Decimal (10 places)

25.2190404258

Rounded

25.2 · 25.22 · 25.219

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.219040

Prime factorization

2² × 3 × 53

Cube root

8.599748

How to simplify √636

Look for the largest perfect square that divides 636. Here it is 4 (2²), because 636 = 4 × 159 and 159 has no square factor left:

The prime factorization tells the same story: 636 = 2² × 3 × 53. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 53 stays inside.

Check: (2√159)² = 2² × 159 = 4 × 159 = 636. As a decimal, 2√159 = 2 × 12.6095202129 ≈ 25.2190404258.

Where √636 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √636 lies between 25 and 26. 636 is 11 above 625 and 40 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.2157 (0.01% low)
• Tangent from 25, i.e. 25 + 11 ÷ 50: 25.2200 (0% high)
• Tangent from 26, i.e. 26 − 40 ÷ 52: 25.2308 (0.05% high)

For √636 the tangent at 25 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 636 is just 11 above 625.

Finding √636 with the Babylonian method

Picture a rectangle with an area of 636 and one side x; the other side must be 636 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √636.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √636 = 25.2190404258 to every decimal shown.

√636 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √636 the pattern is [25; 4, 1, 1, 3, 3, 12, 3, 3, 1, 1, 4, 50] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √636 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 − 636y² = 1. Its smallest solution in positive whole numbers is x = 3,505,951, y = 139,020.

√636 in geometry and everyday measurements

• A square garage floor of 636 square feet measures about 25.22 ft (25 ft 3 in) per side, and its corner-to-corner diagonal is √1272 ≈ 35.7 ft.
• 636 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 √636 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 √636 as its space diagonal.
• Since √636 = 2√159, a length of √636 is exactly 2 copies of the length √159 laid end to end.

Square roots near √636 and related roots

• The cube root of 636 is about 8.599748.
• Because 636 = 4 × 159, the root is twice √159: 2 × 12.60952 ≈ 25.21904.

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