Square Root of 561

√561 at a glance

Exact value

√561

Decimal (10 places)

23.6854385647

Rounded

23.7 · 23.69 · 23.685

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.685439

Prime factorization

3 × 11 × 17

Cube root

8.247474

How to simplify √561

The prime factorization of 561 is 3 × 11 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √561 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 561, 3, 11 and 17 appear an odd number of times, so √561 is irrational and 23.6854385647 is a rounded value.

Where √561 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √561 lies between 23 and 24. 561 is 32 above 529 and 15 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.6809 (0.02% low)
• Tangent from 23, i.e. 23 + 32 ÷ 46: 23.6957 (0.04% high)
• Tangent from 24, i.e. 24 − 15 ÷ 48: 23.6875 (0.01% high)

For √561 the tangent at 24 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 561 is just 15 below 576.

Finding √561 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 561: following the tangent line down to zero simplifies to averaging x with 561 ÷ x.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √561 = 23.6854385647 to every decimal shown.

√561 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √561 the pattern is [23; 1, 2, 5, 1, 1, 2, 2, 2, 1, 1, 5, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √561 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 − 561y² = 1. Its smallest solution in positive whole numbers is x = 522,785, y = 22,072.

√561 in geometry and everyday measurements

• A square garage floor of 561 square feet measures about 23.69 ft (23 ft 8 in) per side, and its corner-to-corner diagonal is √1122 ≈ 33.5 ft.
• 561 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √561 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 14 × 19 box, because 2² + 14² + 19² = 561.

Square roots near √561 and related roots

• The cube root of 561 is about 8.247474.
• Squaring undoes the root: (√561)² = 561, while 561² = 314,721 — the number whose square root is 561.

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