Square Root of 632

√632 at a glance

Exact value

2√158

Decimal (10 places)

25.1396101800

Rounded

25.1 · 25.14 · 25.140

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.139610

Prime factorization

2³ × 79

Cube root

8.581681

How to simplify √632

Look for the largest perfect square that divides 632. Here it is 4 (2²), because 632 = 4 × 158 and 158 has no square factor left:

The prime factorization tells the same story: 632 = 2³ × 79. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 79 stays inside.

Check: (2√158)² = 2² × 158 = 4 × 158 = 632. As a decimal, 2√158 = 2 × 12.56980509 ≈ 25.1396101800.

Where √632 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √632 lies between 25 and 26. 632 is 7 above 625 and 44 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.1373 (0.01% low)
• Tangent from 25, i.e. 25 + 7 ÷ 50: 25.1400 (0% high)
• Tangent from 26, i.e. 26 − 44 ÷ 52: 25.1538 (0.06% high)

For √632 the tangent at 25 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 632 is just 7 above 625.

Finding √632 with the Babylonian method

Picture a rectangle with an area of 632 and one side x; the other side must be 632 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √632.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √632 = 25.1396101800 to every decimal shown.

√632 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √632 the pattern is [25; 7, 6, 7, 50] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √632 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 − 632y² = 1. Its smallest solution in positive whole numbers is x = 7,743, y = 308.

√632 in geometry and everyday measurements

• A square garage floor of 632 square feet measures about 25.14 ft (25 ft 2 in) per side, and its corner-to-corner diagonal is √1264 ≈ 35.6 ft.
• 632 is not a sum of two whole-number squares — the prime factor 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √632 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 12 × 22 box, because 2² + 12² + 22² = 632.
• Since √632 = 2√158, a length of √632 is exactly 2 copies of the length √158 laid end to end.

Square roots near √632 and related roots

• The cube root of 632 is about 8.581681.
• Because 632 = 4 × 158, the root is twice √158: 2 × 12.569805 ≈ 25.13961.

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