Square Root of 651

√651 at a glance

Exact value

√651

Decimal (10 places)

25.5147016443

Rounded

25.5 · 25.51 · 25.515

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.514702

Prime factorization

3 × 7 × 31

Cube root

8.666831

How to simplify √651

The prime factorization of 651 is 3 × 7 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √651 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 651, 3, 7 and 31 appear an odd number of times, so √651 is irrational and 25.5147016443 is a rounded value.

Where √651 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √651 lies between 25 and 26. 651 is 26 above 625 and 25 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.5098 (0.02% low)
• Tangent from 25, i.e. 25 + 26 ÷ 50: 25.5200 (0.02% high)
• Tangent from 26, i.e. 26 − 25 ÷ 52: 25.5192 (0.02% high)

For √651 the tangent at 26 wins, missing by only 0.0045. Tangent estimates shine when the number sits close to a perfect square — here 651 is just 25 below 676.

Finding √651 with the Babylonian method

If a guess is too big, 651 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√651) in one step.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √651 = 25.5147016443 to every decimal shown.

√651 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √651 the pattern is [25; 1, 1, 16, 1, 1, 50] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √651 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 − 651y² = 1. Its smallest solution in positive whole numbers is x = 1,735, y = 68.

√651 in geometry and everyday measurements

• 651 square feet is 60.5 m². Laid out as a square — a small house footprint or a lot — it is about 25.51 ft (25 ft 6 in) on a side.
• 651 is not a sum of two whole-number squares — the prime factor 3, 7 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √651 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 25 box, because 1² + 5² + 25² = 651.

Square roots near √651 and related roots

• The cube root of 651 is about 8.666831.
• Squaring undoes the root: (√651)² = 651, while 651² = 423,801 — the number whose square root is 651.

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