Square Root of 880

√880 at a glance

Exact value

4√55

Decimal (10 places)

29.6647939484

Rounded

29.7 · 29.66 · 29.665

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.664794

Prime factorization

2⁴ × 5 × 11

Cube root

9.582840

How to simplify √880

Look for the largest perfect square that divides 880. Here it is 16 (4²), because 880 = 16 × 55 and 55 has no square factor left:

The prime factorization tells the same story: 880 = 2⁴ × 5 × 11. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 5 × 11 stays inside.

880 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √880 = 2√220, and √220 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√55)² = 4² × 55 = 16 × 55 = 880. As a decimal, 4√55 = 4 × 7.4161984871 ≈ 29.6647939484.

Where √880 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √880 lies between 29 and 30. 880 is 39 above 841 and 20 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.6610 (0.01% low)
• Tangent from 29, i.e. 29 + 39 ÷ 58: 29.6724 (0.03% high)
• Tangent from 30, i.e. 30 − 20 ÷ 60: 29.6667 (0.01% high)

For √880 the tangent at 30 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 880 is just 20 below 900.

Finding √880 with the Babylonian method

Picture a rectangle with an area of 880 and one side x; the other side must be 880 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √880.

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 √880 = 29.6647939484 to every decimal shown.

√880 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √880 the pattern is [29; 1, 1, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √880 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 − 880y² = 1. Its smallest solution in positive whole numbers is x = 89, y = 3.

√880 in geometry and everyday measurements

• 880 square feet is 81.8 m². Laid out as a square — a small house footprint or a lot — it is about 29.66 ft (29 ft 8 in) on a side.
• 880 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √880 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 √880 as its space diagonal.
• Since √880 = 4√55, a length of √880 is exactly 4 copies of the length √55 laid end to end.

Square roots near √880 and related roots

• The cube root of 880 is about 9.582840.
• Because 880 = 4 × 220, the root is twice √220: 2 × 14.832397 ≈ 29.664794.

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