Square Root of 212

√212 at a glance

Exact value

2√53

Decimal (10 places)

14.5602197786

Rounded

14.6 · 14.56 · 14.560

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.560220

Prime factorization

2² × 53

Cube root

5.962732

How to simplify √212

Look for the largest perfect square that divides 212. Here it is 4 (2²), because 212 = 4 × 53 and 53 has no square factor left:

The prime factorization tells the same story: 212 = 2² × 53. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 53 stays inside.

Check: (2√53)² = 2² × 53 = 4 × 53 = 212. As a decimal, 2√53 = 2 × 7.2801098893 ≈ 14.5602197786.

Where √212 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √212 lies between 14 and 15. 212 is 16 above 196 and 13 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.5517 (0.06% low)
• Tangent from 14, i.e. 14 + 16 ÷ 28: 14.5714 (0.08% high)
• Tangent from 15, i.e. 15 − 13 ÷ 30: 14.5667 (0.04% high)

For √212 the tangent at 15 wins, missing by only 0.0064. Tangent estimates shine when the number sits close to a perfect square — here 212 is just 13 below 225.

Finding √212 with the Babylonian method

Picture a rectangle with an area of 212 and one side x; the other side must be 212 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √212.

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

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √212 = 14.5602197786 to every decimal shown.

√212 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √212 the pattern is [14; 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √212 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 − 212y² = 1. Its smallest solution in positive whole numbers is x = 66,249, y = 4,550.

√212 in geometry and everyday measurements

• A square patio or deck of 212 square feet is about 14.56 ft (14 ft 7 in) on each side, so edging all the way around takes 4 × √212 ≈ 58.2 ft.
• 212 = 4² + 14², so by the Pythagorean theorem √212 is the diagonal of a 4 × 14 rectangle — and the distance between the points (0, 0) and (4, 14) on a grid.
• Since √212 = 2√53, a length of √212 is exactly 2 copies of the length √53 laid end to end.

Square roots near √212 and related roots

• The cube root of 212 is about 5.962732.
• Four times the radicand doubles the root: √848 = 2 × √212 ≈ 29.12044.

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