Square Root of 224

√224 at a glance

Exact value

4√14

Decimal (10 places)

14.9666295471

Rounded

15.0 · 14.97 · 14.967

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.966630

Prime factorization

2⁵ × 7

Cube root

6.073178

How to simplify √224

Look for the largest perfect square that divides 224. Here it is 16 (4²), because 224 = 16 × 14 and 14 has no square factor left:

The prime factorization tells the same story: 224 = 2⁵ × 7. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 7 stays inside.

224 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √224 = 2√56, and √56 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√14)² = 4² × 14 = 16 × 14 = 224. As a decimal, 4√14 = 4 × 3.7416573868 ≈ 14.9666295471.

Where √224 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √224 lies between 14 and 15. 224 is 28 above 196 and 1 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.9655 (0.01% low)
• Tangent from 14, i.e. 14 + 28 ÷ 28: 15.0000 (0.22% high)
• Tangent from 15, i.e. 15 − 1 ÷ 30: 14.9667 (0% high)

For √224 the tangent at 15 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 224 is just 1 below 225.

Finding √224 with the Babylonian method

Picture a rectangle with an area of 224 and one side x; the other side must be 224 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √224.

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

Because the starting guess was already close, two steps are enough to match √224 = 14.9666295471 to every decimal shown.

√224 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √224 the pattern is [14; 1, 28] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √224 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 − 224y² = 1. Its smallest solution in positive whole numbers is x = 15, y = 1.

√224 in geometry and everyday measurements

• A square patio or deck of 224 square feet is about 14.97 ft (15 ft) on each side, so edging all the way around takes 4 × √224 ≈ 59.9 ft.
• 224 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √224 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 12 box, because 4² + 8² + 12² = 224.
• Since √224 = 4√14, a length of √224 is exactly 4 copies of the length √14 laid end to end.

Square roots near √224 and related roots

• The cube root of 224 is about 6.073178.
• Four times the radicand doubles the root: √896 = 2 × √224 ≈ 29.933259.

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