Square Root of 892

√892 at a glance

Exact value

2√223

Decimal (10 places)

29.8663690461

Rounded

29.9 · 29.87 · 29.866

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.866369

Prime factorization

2² × 223

Cube root

9.626202

How to simplify √892

Look for the largest perfect square that divides 892. Here it is 4 (2²), because 892 = 4 × 223 and 223 has no square factor left:

The prime factorization tells the same story: 892 = 2² × 223. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 223 stays inside.

Check: (2√223)² = 2² × 223 = 4 × 223 = 892. As a decimal, 2√223 = 2 × 14.9331845231 ≈ 29.8663690461.

Where √892 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √892 lies between 29 and 30. 892 is 51 above 841 and 8 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.8644 (0.01% low)
• Tangent from 29, i.e. 29 + 51 ÷ 58: 29.8793 (0.04% high)
• Tangent from 30, i.e. 30 − 8 ÷ 60: 29.8667 (0% high)

For √892 the tangent at 30 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 892 is just 8 below 900.

Finding √892 with the Babylonian method

Picture a rectangle with an area of 892 and one side x; the other side must be 892 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √892.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √892 = 29.8663690461 to every decimal shown.

√892 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √892 the pattern is [29; 1, 6, 2, 14, 2, 6, 1, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √892 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 − 892y² = 1. Its smallest solution in positive whole numbers is x = 100,351, y = 3,360.

√892 in geometry and everyday measurements

• 892 square feet is 82.9 m². Laid out as a square — a small house footprint or a lot — it is about 29.87 ft (29 ft 10 in) on a side.
• 892 is not a sum of two whole-number squares — the prime factor 223 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √892 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 √892 as its space diagonal.
• Since √892 = 2√223, a length of √892 is exactly 2 copies of the length √223 laid end to end.

Square roots near √892 and related roots

• The cube root of 892 is about 9.626202.
• Because 892 = 4 × 223, the root is twice √223: 2 × 14.933185 ≈ 29.866369.

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