Square Root of 654

√654 at a glance

Exact value

√654

Decimal (10 places)

25.5734237051

Rounded

25.6 · 25.57 · 25.573

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.573424

Prime factorization

2 × 3 × 109

Cube root

8.680124

How to simplify √654

The prime factorization of 654 is 2 × 3 × 109. Every prime appears only once, so there is no pair to bring outside the radical — √654 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 654, 2, 3 and 109 appear an odd number of times, so √654 is irrational and 25.5734237051 is a rounded value.

Where √654 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √654 lies between 25 and 26. 654 is 29 above 625 and 22 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.5686 (0.02% low)
• Tangent from 25, i.e. 25 + 29 ÷ 50: 25.5800 (0.03% high)
• Tangent from 26, i.e. 26 − 22 ÷ 52: 25.5769 (0.01% high)

For √654 the tangent at 26 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 654 is just 22 below 676.

Finding √654 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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 √654 = 25.5734237051 to every decimal shown.

√654 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √654 the pattern is [25; 1, 1, 2, 1, 9, 1, 1, 16, 1, 1, 9, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √654 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 − 654y² = 1. Its smallest solution in positive whole numbers is x = 8,915,765, y = 348,634.

√654 in geometry and everyday measurements

• 654 square feet is 60.8 m². Laid out as a square — a small house footprint or a lot — it is about 25.57 ft (25 ft 7 in) on a side.
• 654 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 √654 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 13 × 22 box, because 1² + 13² + 22² = 654.

Square roots near √654 and related roots

• The cube root of 654 is about 8.680124.
• Squaring undoes the root: (√654)² = 654, while 654² = 427,716 — the number whose square root is 654.

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