Square Root of 865

√865 at a glance

Exact value

√865

Decimal (10 places)

29.4108823397

Rounded

29.4 · 29.41 · 29.411

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.410882

Prime factorization

5 × 173

Cube root

9.528079

How to simplify √865

The prime factorization of 865 is 5 × 173. Every prime appears only once, so there is no pair to bring outside the radical — √865 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 865, 5 and 173 appear an odd number of times, so √865 is irrational and 29.4108823397 is a rounded value.

Where √865 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √865 lies between 29 and 30. 865 is 24 above 841 and 35 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.4068 (0.01% low)
• Tangent from 29, i.e. 29 + 24 ÷ 58: 29.4138 (0.01% high)
• Tangent from 30, i.e. 30 − 35 ÷ 60: 29.4167 (0.02% high)

For √865 the tangent at 29 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 865 is just 24 above 841.

Finding √865 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 865: following the tangent line down to zero simplifies to averaging x with 865 ÷ x.

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 √865 = 29.4108823397 to every decimal shown.

√865 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √865 the pattern is [29; 2, 2, 3, 3, 1, 1, 1, 2, 6, 6, 2, 1, …] with the block of 19 terms after the semicolon repeating forever (only the first 12 of the 19 are shown). A pattern that never ends is one more proof that √865 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 − 865y² = 1. Its smallest solution in positive whole numbers is x = 242,688,628,535,063,329, y = 8,251,660,923,733,224 — 18 digits for x, even though 865 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 348,345,108² − 865 × 11,844,089² = −1.

√865 in geometry and everyday measurements

• 865 square feet is 80.4 m². Laid out as a square — a small house footprint or a lot — it is about 29.41 ft (29 ft 5 in) on a side.
• 865 = 9² + 28² = 17² + 24², so by the Pythagorean theorem √865 is the diagonal of rectangles measuring 9 × 28 and 17 × 24 — and the distance between the points (0, 0) and (9, 28) on a grid.

Square roots near √865 and related roots

• The cube root of 865 is about 9.528079.
• Squaring undoes the root: (√865)² = 865, while 865² = 748,225 — the number whose square root is 865.

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