Square Root of 220

√220 at a glance

Exact value

2√55

Decimal (10 places)

14.8323969742

Rounded

14.8 · 14.83 · 14.832

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.832397

Prime factorization

2² × 5 × 11

Cube root

6.036811

How to simplify √220

Look for the largest perfect square that divides 220. Here it is 4 (2²), because 220 = 4 × 55 and 55 has no square factor left:

The prime factorization tells the same story: 220 = 2² × 5 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 11 stays inside.

Check: (2√55)² = 2² × 55 = 4 × 55 = 220. As a decimal, 2√55 = 2 × 7.4161984871 ≈ 14.8323969742.

Where √220 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √220 lies between 14 and 15. 220 is 24 above 196 and 5 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.8276 (0.03% low)
• Tangent from 14, i.e. 14 + 24 ÷ 28: 14.8571 (0.17% high)
• Tangent from 15, i.e. 15 − 5 ÷ 30: 14.8333 (0.01% high)

For √220 the tangent at 15 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 220 is just 5 below 225.

Finding √220 with the Babylonian method

Picture a rectangle with an area of 220 and one side x; the other side must be 220 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √220.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √220 = 14.8323969742 to every decimal shown.

√220 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √220 the pattern is [14; 1, 4, 1, 28] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √220 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 − 220y² = 1. Its smallest solution in positive whole numbers is x = 89, y = 6.

√220 in geometry and everyday measurements

• A square patio or deck of 220 square feet is about 14.83 ft (14 ft 10 in) on each side, so edging all the way around takes 4 × √220 ≈ 59.3 ft.
• 220 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 √220 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 √220 as its space diagonal.
• Since √220 = 2√55, a length of √220 is exactly 2 copies of the length √55 laid end to end.

Square roots near √220 and related roots

• The cube root of 220 is about 6.036811.
• Four times the radicand doubles the root: √880 = 2 × √220 ≈ 29.664794.

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