Square Root of 228

√228 at a glance

Exact value

2√57

Decimal (10 places)

15.0996688705

Rounded

15.1 · 15.10 · 15.100

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.099669

Prime factorization

2² × 3 × 19

Cube root

6.109115

How to simplify √228

Look for the largest perfect square that divides 228. Here it is 4 (2²), because 228 = 4 × 57 and 57 has no square factor left:

The prime factorization tells the same story: 228 = 2² × 3 × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 19 stays inside.

Check: (2√57)² = 2² × 57 = 4 × 57 = 228. As a decimal, 2√57 = 2 × 7.5498344353 ≈ 15.0996688705.

Where √228 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √228 lies between 15 and 16. 228 is 3 above 225 and 28 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.0968 (0.02% low)
• Tangent from 15, i.e. 15 + 3 ÷ 30: 15.1000 (0% high)
• Tangent from 16, i.e. 16 − 28 ÷ 32: 15.1250 (0.17% high)

For √228 the tangent at 15 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 228 is just 3 above 225.

Finding √228 with the Babylonian method

Picture a rectangle with an area of 228 and one side x; the other side must be 228 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √228.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √228 = 15.0996688705 to every decimal shown.

√228 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √228 the pattern is [15; 10, 30] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √228 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 − 228y² = 1. Its smallest solution in positive whole numbers is x = 151, y = 10.

√228 in geometry and everyday measurements

• A square patio or deck of 228 square feet is about 15.1 ft (15 ft 1 in) on each side, so edging all the way around takes 4 × √228 ≈ 60.4 ft.
• 228 is not a sum of two whole-number squares — the prime factor 3 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √228 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 14 box, because 4² + 4² + 14² = 228.
• Since √228 = 2√57, a length of √228 is exactly 2 copies of the length √57 laid end to end.

Square roots near √228 and related roots

• The cube root of 228 is about 6.109115.
• Four times the radicand doubles the root: √912 = 2 × √228 ≈ 30.199338.

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