Square Root of 872

√872 at a glance

Exact value

2√218

Decimal (10 places)

29.5296461205

Rounded

29.5 · 29.53 · 29.530

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.529646

Prime factorization

2³ × 109

Cube root

9.553712

How to simplify √872

Look for the largest perfect square that divides 872. Here it is 4 (2²), because 872 = 4 × 218 and 218 has no square factor left:

The prime factorization tells the same story: 872 = 2³ × 109. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 109 stays inside.

Check: (2√218)² = 2² × 218 = 4 × 218 = 872. As a decimal, 2√218 = 2 × 14.7648230602 ≈ 29.5296461205.

Where √872 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √872 lies between 29 and 30. 872 is 31 above 841 and 28 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.5254 (0.01% low)
• Tangent from 29, i.e. 29 + 31 ÷ 58: 29.5345 (0.02% high)
• Tangent from 30, i.e. 30 − 28 ÷ 60: 29.5333 (0.01% high)

For √872 the tangent at 30 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 872 is just 28 below 900.

Finding √872 with the Babylonian method

Picture a rectangle with an area of 872 and one side x; the other side must be 872 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √872.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √872 = 29.5296461205 to every decimal shown.

√872 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √872 the pattern is [29; 1, 1, 7, 1, 13, 1, 7, 1, 1, 58] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √872 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 − 872y² = 1. Its smallest solution in positive whole numbers is x = 126,003, y = 4,267.

√872 in geometry and everyday measurements

• 872 square feet is 81 m². Laid out as a square — a small house footprint or a lot — it is about 29.53 ft (29 ft 6 in) on a side.
• 872 = 14² + 26², so by the Pythagorean theorem √872 is the diagonal of a 14 × 26 rectangle — and the distance between the points (0, 0) and (14, 26) on a grid.
• Since √872 = 2√218, a length of √872 is exactly 2 copies of the length √218 laid end to end.

Square roots near √872 and related roots

• The cube root of 872 is about 9.553712.
• Because 872 = 4 × 218, the root is twice √218: 2 × 14.764823 ≈ 29.529646.

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