Square Root of 216

√216 at a glance

Exact value

6√6

Decimal (10 places)

14.6969384567

Rounded

14.7 · 14.70 · 14.697

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.696938

Prime factorization

2³ × 3³

Cube root

6

How to simplify √216

Look for the largest perfect square that divides 216. Here it is 36 (6²), because 216 = 36 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 216 = 2³ × 3³. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 2 × 3 stays inside.

216 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √216 = 2√54, and √54 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√6)² = 6² × 6 = 36 × 6 = 216. As a decimal, 6√6 = 6 × 2.4494897428 ≈ 14.6969384567.

Where √216 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √216 lies between 14 and 15. 216 is 20 above 196 and 9 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.6897 (0.05% low)
• Tangent from 14, i.e. 14 + 20 ÷ 28: 14.7143 (0.12% high)
• Tangent from 15, i.e. 15 − 9 ÷ 30: 14.7000 (0.02% high)

For √216 the tangent at 15 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 216 is just 9 below 225.

Finding √216 with the Babylonian method

Picture a rectangle with an area of 216 and one side x; the other side must be 216 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √216.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √216 = 14.6969384567 to every decimal shown.

√216 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √216 the pattern is [14; 1, 2, 3, 2, 1, 28] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √216 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 − 216y² = 1. Its smallest solution in positive whole numbers is x = 485, y = 33.

√216 in geometry and everyday measurements

• A square patio or deck of 216 square feet is about 14.7 ft (14 ft 8 in) on each side, so edging all the way around takes 4 × √216 ≈ 58.8 ft.
• 216 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √216 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 14 box, because 2² + 4² + 14² = 216.
• Since √216 = 6√6, a length of √216 is exactly 6 copies of the length √6 laid end to end.

Square roots near √216 and related roots

• The cube root of 216 is exactly 6 — 216 is a perfect cube as well (6³).
• Four times the radicand doubles the root: √864 = 2 × √216 ≈ 29.393877.

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