Square Root of 884

√884 at a glance

Exact value

2√221

Decimal (10 places)

29.7321374946

Rounded

29.7 · 29.73 · 29.732

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.732137

Prime factorization

2² × 13 × 17

Cube root

9.597337

How to simplify √884

Look for the largest perfect square that divides 884. Here it is 4 (2²), because 884 = 4 × 221 and 221 has no square factor left:

The prime factorization tells the same story: 884 = 2² × 13 × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 13 × 17 stays inside.

Check: (2√221)² = 2² × 221 = 4 × 221 = 884. As a decimal, 2√221 = 2 × 14.8660687473 ≈ 29.7321374946.

Where √884 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √884 lies between 29 and 30. 884 is 43 above 841 and 16 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.7288 (0.01% low)
• Tangent from 29, i.e. 29 + 43 ÷ 58: 29.7414 (0.03% high)
• Tangent from 30, i.e. 30 − 16 ÷ 60: 29.7333 (0% high)

For √884 the tangent at 30 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 884 is just 16 below 900.

Finding √884 with the Babylonian method

Picture a rectangle with an area of 884 and one side x; the other side must be 884 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √884.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √884 = 29.7321374946 to every decimal shown.

√884 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √884 the pattern is [29; 1, 2, 1, 2, 1, 2, 1, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √884 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 − 884y² = 1. Its smallest solution in positive whole numbers is x = 1,665, y = 56.

√884 in geometry and everyday measurements

• 884 square feet is 82.1 m². Laid out as a square — a small house footprint or a lot — it is about 29.73 ft (29 ft 9 in) on a side.
• 884 = 10² + 28² = 20² + 22², so by the Pythagorean theorem √884 is the diagonal of rectangles measuring 10 × 28 and 20 × 22 — and the distance between the points (0, 0) and (10, 28) on a grid.
• Since √884 = 2√221, a length of √884 is exactly 2 copies of the length √221 laid end to end.

Square roots near √884 and related roots

• The cube root of 884 is about 9.597337.
• Because 884 = 4 × 221, the root is twice √221: 2 × 14.866069 ≈ 29.732137.

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