Square Root of 861

√861 at a glance

Exact value

√861

Decimal (10 places)

29.3428015022

Rounded

29.3 · 29.34 · 29.343

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.342802

Prime factorization

3 × 7 × 41

Cube root

9.513370

How to simplify √861

The prime factorization of 861 is 3 × 7 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √861 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 861, 3, 7 and 41 appear an odd number of times, so √861 is irrational and 29.3428015022 is a rounded value.

Where √861 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √861 lies between 29 and 30. 861 is 20 above 841 and 39 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.3390 (0.01% low)
• Tangent from 29, i.e. 29 + 20 ÷ 58: 29.3448 (0.01% high)
• Tangent from 30, i.e. 30 − 39 ÷ 60: 29.3500 (0.02% high)

For √861 the tangent at 29 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 861 is just 20 above 841.

Finding √861 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 861: following the tangent line down to zero simplifies to averaging x with 861 ÷ x.

Start from the nearest whole number, 29 (29² = 841):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √861 = 29.3428015022 to every decimal shown.

√861 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √861 the pattern is [29; 2, 1, 11, 14, 1, 1, 2, 2, 2, 1, 1, 14, …] 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 √861 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 − 861y² = 1. Its smallest solution in positive whole numbers is x = 541,601,801, y = 18,457,740.

√861 in geometry and everyday measurements

• 861 square feet is 80 m². Laid out as a square — a small house footprint or a lot — it is about 29.34 ft (29 ft 4 in) on a side.
• 861 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √861 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 29 box, because 2² + 4² + 29² = 861.

Square roots near √861 and related roots

• The cube root of 861 is about 9.513370.
• Squaring undoes the root: (√861)² = 861, while 861² = 741,321 — the number whose square root is 861.

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