Square Root of 440

√440 at a glance

Exact value

2√110

Decimal (10 places)

20.9761769634

Rounded

21.0 · 20.98 · 20.976

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.976177

Prime factorization

2³ × 5 × 11

Cube root

7.605905

How to simplify √440

Look for the largest perfect square that divides 440. Here it is 4 (2²), because 440 = 4 × 110 and 110 has no square factor left:

The prime factorization tells the same story: 440 = 2³ × 5 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 5 × 11 stays inside.

Check: (2√110)² = 2² × 110 = 4 × 110 = 440. As a decimal, 2√110 = 2 × 10.4880884817 ≈ 20.9761769634.

Where √440 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √440 lies between 20 and 21. 440 is 40 above 400 and 1 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.9756 (0% low)
• Tangent from 20, i.e. 20 + 40 ÷ 40: 21.0000 (0.11% high)
• Tangent from 21, i.e. 21 − 1 ÷ 42: 20.9762 (0% high)

For √440 the tangent at 21 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 440 is just 1 below 441.

Finding √440 with the Babylonian method

Picture a rectangle with an area of 440 and one side x; the other side must be 440 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √440.

Start from the nearest whole number, 21 (21² = 441):

Because the starting guess was already close, two steps are enough to match √440 = 20.9761769634 to every decimal shown.

√440 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √440 the pattern is [20; 1, 40] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √440 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 − 440y² = 1. Its smallest solution in positive whole numbers is x = 21, y = 1.

√440 in geometry and everyday measurements

• A square garage floor of 440 square feet measures about 20.98 ft (21 ft) per side, and its corner-to-corner diagonal is √880 ≈ 29.7 ft.
• 440 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 √440 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 20 box, because 2² + 6² + 20² = 440.
• Since √440 = 2√110, a length of √440 is exactly 2 copies of the length √110 laid end to end.

Square roots near √440 and related roots

• The cube root of 440 is about 7.605905.
• Because 440 = 4 × 110, the root is twice √110: 2 × 10.488088 ≈ 20.976177.

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