Square Root of 860

√860 at a glance

Exact value

2√215

Decimal (10 places)

29.3257565972

Rounded

29.3 · 29.33 · 29.326

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.325757

Prime factorization

2² × 5 × 43

Cube root

9.509685

How to simplify √860

Look for the largest perfect square that divides 860. Here it is 4 (2²), because 860 = 4 × 215 and 215 has no square factor left:

The prime factorization tells the same story: 860 = 2² × 5 × 43. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 43 stays inside.

Check: (2√215)² = 2² × 215 = 4 × 215 = 860. As a decimal, 2√215 = 2 × 14.6628782986 ≈ 29.3257565972.

Where √860 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √860 lies between 29 and 30. 860 is 19 above 841 and 40 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.3220 (0.01% low)
• Tangent from 29, i.e. 29 + 19 ÷ 58: 29.3276 (0.01% high)
• Tangent from 30, i.e. 30 − 40 ÷ 60: 29.3333 (0.03% high)

For √860 the tangent at 29 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 860 is just 19 above 841.

Finding √860 with the Babylonian method

Picture a rectangle with an area of 860 and one side x; the other side must be 860 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √860.

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 √860 = 29.3257565972 to every decimal shown.

√860 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √860 the pattern is [29; 3, 14, 3, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √860 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 − 860y² = 1. Its smallest solution in positive whole numbers is x = 3,871, y = 132.

√860 in geometry and everyday measurements

• 860 square feet is 79.9 m². Laid out as a square — a small house footprint or a lot — it is about 29.33 ft (29 ft 4 in) on a side.
• 860 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √860 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √860 as its space diagonal.
• Since √860 = 2√215, a length of √860 is exactly 2 copies of the length √215 laid end to end.

Square roots near √860 and related roots

• The cube root of 860 is about 9.509685.
• Because 860 = 4 × 215, the root is twice √215: 2 × 14.662878 ≈ 29.325757.

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