Square Root of 876

√876 at a glance

Exact value

2√219

Decimal (10 places)

29.5972971739

Rounded

29.6 · 29.60 · 29.597

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.597297

Prime factorization

2² × 3 × 73

Cube root

9.568298

How to simplify √876

Look for the largest perfect square that divides 876. Here it is 4 (2²), because 876 = 4 × 219 and 219 has no square factor left:

The prime factorization tells the same story: 876 = 2² × 3 × 73. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 73 stays inside.

Check: (2√219)² = 2² × 219 = 4 × 219 = 876. As a decimal, 2√219 = 2 × 14.7986485869 ≈ 29.5972971739.

Where √876 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √876 lies between 29 and 30. 876 is 35 above 841 and 24 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.5932 (0.01% low)
• Tangent from 29, i.e. 29 + 35 ÷ 58: 29.6034 (0.02% high)
• Tangent from 30, i.e. 30 − 24 ÷ 60: 29.6000 (0.01% high)

For √876 the tangent at 30 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 876 is just 24 below 900.

Finding √876 with the Babylonian method

Picture a rectangle with an area of 876 and one side x; the other side must be 876 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √876.

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 √876 = 29.5972971739 to every decimal shown.

√876 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √876 the pattern is [29; 1, 1, 2, 14, 2, 1, 1, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √876 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 − 876y² = 1. Its smallest solution in positive whole numbers is x = 10,951, y = 370.

√876 in geometry and everyday measurements

• 876 square feet is 81.4 m². Laid out as a square — a small house footprint or a lot — it is about 29.6 ft (29 ft 7 in) on a side.
• 876 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √876 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 14 × 26 box, because 2² + 14² + 26² = 876.
• Since √876 = 2√219, a length of √876 is exactly 2 copies of the length √219 laid end to end.

Square roots near √876 and related roots

• The cube root of 876 is about 9.568298.
• Because 876 = 4 × 219, the root is twice √219: 2 × 14.798649 ≈ 29.597297.

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