Square Root of 866

√866 at a glance

Exact value

√866

Decimal (10 places)

29.4278779391

Rounded

29.4 · 29.43 · 29.428

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.427878

Prime factorization

2 × 433

Cube root

9.531750

How to simplify √866

The prime factorization of 866 is 2 × 433. Every prime appears only once, so there is no pair to bring outside the radical — √866 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 866, 2 and 433 appear an odd number of times, so √866 is irrational and 29.4278779391 is a rounded value.

Where √866 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √866 lies between 29 and 30. 866 is 25 above 841 and 34 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.4237 (0.01% low)
• Tangent from 29, i.e. 29 + 25 ÷ 58: 29.4310 (0.01% high)
• Tangent from 30, i.e. 30 − 34 ÷ 60: 29.4333 (0.02% high)

For √866 the tangent at 29 wins, missing by only 0.0032. Tangent estimates shine when the number sits close to a perfect square — here 866 is just 25 above 841.

Finding √866 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, 29 (29² = 841):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √866 = 29.4278779391 to every decimal shown.

√866 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √866 the pattern is [29; 2, 2, 1, 28, 1, 2, 2, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √866 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 − 866y² = 1. Its smallest solution in positive whole numbers is x = 42,435, y = 1,442.

√866 in geometry and everyday measurements

• 866 square feet is 80.5 m². Laid out as a square — a small house footprint or a lot — it is about 29.43 ft (29 ft 5 in) on a side.
• 866 = 5² + 29², so by the Pythagorean theorem √866 is the diagonal of a 5 × 29 rectangle — and the distance between the points (0, 0) and (5, 29) on a grid.

Square roots near √866 and related roots

• The cube root of 866 is about 9.531750.
• Squaring undoes the root: (√866)² = 866, while 866² = 749,956 — the number whose square root is 866.

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