Square Root of 550

√550 at a glance

Exact value

5√22

Decimal (10 places)

23.4520787991

Rounded

23.5 · 23.45 · 23.452

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.452079

Prime factorization

2 × 5² × 11

Cube root

8.193213

How to simplify √550

Look for the largest perfect square that divides 550. Here it is 25 (5²), because 550 = 25 × 22 and 22 has no square factor left:

The prime factorization tells the same story: 550 = 2 × 5² × 11. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 11 stays inside.

Check: (5√22)² = 5² × 22 = 25 × 22 = 550. As a decimal, 5√22 = 5 × 4.6904157598 ≈ 23.4520787991.

Where √550 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √550 lies between 23 and 24. 550 is 21 above 529 and 26 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.4468 (0.02% low)
• Tangent from 23, i.e. 23 + 21 ÷ 46: 23.4565 (0.02% high)
• Tangent from 24, i.e. 24 − 26 ÷ 48: 23.4583 (0.03% high)

For √550 the tangent at 23 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 550 is just 21 above 529.

Finding √550 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √550 = 23.4520787991 to every decimal shown.

√550 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √550 the pattern is [23; 2, 4, 1, 2, 1, 1, 7, 4, 7, 1, 1, 2, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √550 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 − 550y² = 1. Its smallest solution in positive whole numbers is x = 30,580,901, y = 1,303,974.

√550 in geometry and everyday measurements

• A square garage floor of 550 square feet measures about 23.45 ft (23 ft 5 in) per side, and its corner-to-corner diagonal is √1100 ≈ 33.2 ft.
• 550 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 √550 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 15 × 18 box, because 1² + 15² + 18² = 550.
• Since √550 = 5√22, a length of √550 is exactly 5 copies of the length √22 laid end to end.

Square roots near √550 and related roots

• The cube root of 550 is about 8.193213.
• Squaring undoes the root: (√550)² = 550, while 550² = 302,500 — the number whose square root is 550.

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