Square Root of 864

√864 at a glance

Exact value

12√6

Decimal (10 places)

29.3938769134

Rounded

29.4 · 29.39 · 29.394

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.393877

Prime factorization

2⁵ × 3³

Cube root

9.524406

How to simplify √864

Look for the largest perfect square that divides 864. Here it is 144 (12²), because 864 = 144 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 864 = 2⁵ × 3³. Each pair of equal primes leaves the radical as one factor, so 2² × 3 comes out and 2 × 3 stays inside.

864 has 5 square factors (4, 9, 16, 36 and 144). Starting with a smaller one still works but takes more rounds: √864 = 2√216, and √216 can be simplified again. Using 144 straight away finishes in one step.

Check: (12√6)² = 12² × 6 = 144 × 6 = 864. As a decimal, 12√6 = 12 × 2.4494897428 ≈ 29.3938769134.

Where √864 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √864 lies between 29 and 30. 864 is 23 above 841 and 36 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.3898 (0.01% low)
• Tangent from 29, i.e. 29 + 23 ÷ 58: 29.3966 (0.01% high)
• Tangent from 30, i.e. 30 − 36 ÷ 60: 29.4000 (0.02% high)

For √864 the tangent at 29 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 864 is just 23 above 841.

Finding √864 with the Babylonian method

Picture a rectangle with an area of 864 and one side x; the other side must be 864 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √864.

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 √864 = 29.3938769134 to every decimal shown.

√864 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √864 the pattern is [29; 2, 1, 1, 5, 1, 13, 1, 5, 1, 1, 2, 58] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √864 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 − 864y² = 1. Its smallest solution in positive whole numbers is x = 470,449, y = 16,005.

√864 in geometry and everyday measurements

• 864 square feet is 80.3 m². Laid out as a square — a small house footprint or a lot — it is about 29.39 ft (29 ft 5 in) on a side.
• 864 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 √864 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 28 box, because 4² + 8² + 28² = 864.
• Since √864 = 12√6, a length of √864 is exactly 12 copies of the length √6 laid end to end.

Square roots near √864 and related roots

• The cube root of 864 is about 9.524406.
• Because 864 = 4 × 216, the root is twice √216: 2 × 14.696938 ≈ 29.393877.

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