Square Root of 852

√852 at a glance

Exact value

2√213

Decimal (10 places)

29.1890390387

Rounded

29.2 · 29.19 · 29.189

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.189039

Prime factorization

2² × 3 × 71

Cube root

9.480106

How to simplify √852

Look for the largest perfect square that divides 852. Here it is 4 (2²), because 852 = 4 × 213 and 213 has no square factor left:

The prime factorization tells the same story: 852 = 2² × 3 × 71. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 71 stays inside.

Check: (2√213)² = 2² × 213 = 4 × 213 = 852. As a decimal, 2√213 = 2 × 14.5945195193 ≈ 29.1890390387.

Where √852 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √852 lies between 29 and 30. 852 is 11 above 841 and 48 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.1864 (0.01% low)
• Tangent from 29, i.e. 29 + 11 ÷ 58: 29.1897 (0% high)
• Tangent from 30, i.e. 30 − 48 ÷ 60: 29.2000 (0.04% high)

For √852 the tangent at 29 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 852 is just 11 above 841.

Finding √852 with the Babylonian method

Picture a rectangle with an area of 852 and one side x; the other side must be 852 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √852.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √852 = 29.1890390387 to every decimal shown.

√852 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √852 the pattern is [29; 5, 3, 2, 4, 2, 3, 5, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √852 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 − 852y² = 1. Its smallest solution in positive whole numbers is x = 194,399, y = 6,660.

√852 in geometry and everyday measurements

• 852 square feet is 79.2 m². Laid out as a square — a small house footprint or a lot — it is about 29.19 ft (29 ft 2 in) on a side.
• 852 is not a sum of two whole-number squares — the prime factor 3 and 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √852 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 28 box, because 2² + 8² + 28² = 852.
• Since √852 = 2√213, a length of √852 is exactly 2 copies of the length √213 laid end to end.

Square roots near √852 and related roots

• The cube root of 852 is about 9.480106.
• Because 852 = 4 × 213, the root is twice √213: 2 × 14.59452 ≈ 29.189039.

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