Square Root of 663

√663 at a glance

Exact value

√663

Decimal (10 places)

25.7487863792

Rounded

25.7 · 25.75 · 25.749

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.748786

Prime factorization

3 × 13 × 17

Cube root

8.719760

How to simplify √663

The prime factorization of 663 is 3 × 13 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √663 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 663, 3, 13 and 17 appear an odd number of times, so √663 is irrational and 25.7487863792 is a rounded value.

Where √663 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √663 lies between 25 and 26. 663 is 38 above 625 and 13 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.7451 (0.01% low)
• Tangent from 25, i.e. 25 + 38 ÷ 50: 25.7600 (0.04% high)
• Tangent from 26, i.e. 26 − 13 ÷ 52: 25.7500 (0% high)

For √663 the tangent at 26 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 663 is just 13 below 676.

Finding √663 with the Babylonian method

If a guess is too big, 663 ÷ 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√663) in one step.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √663 = 25.7487863792 to every decimal shown.

√663 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √663 the pattern is [25; 1, 2, 1, 50] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √663 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 − 663y² = 1. Its smallest solution in positive whole numbers is x = 103, y = 4.

√663 in geometry and everyday measurements

• 663 square feet is 61.6 m². Laid out as a square — a small house footprint or a lot — it is about 25.75 ft (25 ft 9 in) on a side.
• 663 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 √663 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 √663 as its space diagonal.

Square roots near √663 and related roots

• The cube root of 663 is about 8.719760.
• Squaring undoes the root: (√663)² = 663, while 663² = 439,569 — the number whose square root is 663.

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