Square Root of 223

√223 at a glance

Exact value

√223

Decimal (10 places)

14.9331845231

Rounded

14.9 · 14.93 · 14.933

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.933185

Prime factorization

223

Cube root

6.064127

How to simplify √223

223 is a prime number, so its only factors are 1 and 223. There is no perfect-square factor to pull out, which means √223 is already in its simplest radical form.

The square root of any prime is irrational. If √223 were a fraction a/b in lowest terms, then a² = 223b², so 223 would divide a — and then 223 would divide b too, contradicting “lowest terms.” That is why the decimal 14.9331845231 is only a rounded value.

Where √223 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √223 lies between 14 and 15. 223 is 27 above 196 and 2 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.9310 (0.01% low)
• Tangent from 14, i.e. 14 + 27 ÷ 28: 14.9643 (0.21% high)
• Tangent from 15, i.e. 15 − 2 ÷ 30: 14.9333 (0% high)

For √223 the tangent at 15 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 223 is just 2 below 225.

Finding √223 with the Babylonian method

If a guess is too big, 223 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√223) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √223 = 14.9331845231 to every decimal shown.

√223 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √223 the pattern is [14; 1, 13, 1, 28] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √223 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 − 223y² = 1. Its smallest solution in positive whole numbers is x = 224, y = 15.

√223 in geometry and everyday measurements

• A square patio or deck of 223 square feet is about 14.93 ft (14 ft 11 in) on each side, so edging all the way around takes 4 × √223 ≈ 59.7 ft.
• 223 is not a sum of two whole-number squares — 223 is itself a prime that is one less than a multiple of 4, which rules that out — so √223 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 √223 as its space diagonal.

Square roots near √223 and related roots

• The cube root of 223 is about 6.064127.
• Four times the radicand doubles the root: √892 = 2 × √223 ≈ 29.866369.

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