Square Root of 516

√516 at a glance

Exact value

2√129

Decimal (10 places)

22.7156333832

Rounded

22.7 · 22.72 · 22.716

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.715633

Prime factorization

2² × 3 × 43

Cube root

8.020779

How to simplify √516

Look for the largest perfect square that divides 516. Here it is 4 (2²), because 516 = 4 × 129 and 129 has no square factor left:

The prime factorization tells the same story: 516 = 2² × 3 × 43. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 43 stays inside.

Check: (2√129)² = 2² × 129 = 4 × 129 = 516. As a decimal, 2√129 = 2 × 11.3578166916 ≈ 22.7156333832.

Where √516 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √516 lies between 22 and 23. 516 is 32 above 484 and 13 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.7111 (0.02% low)
• Tangent from 22, i.e. 22 + 32 ÷ 44: 22.7273 (0.05% high)
• Tangent from 23, i.e. 23 − 13 ÷ 46: 22.7174 (0.01% high)

For √516 the tangent at 23 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 516 is just 13 below 529.

Finding √516 with the Babylonian method

Picture a rectangle with an area of 516 and one side x; the other side must be 516 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √516.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √516 = 22.7156333832 to every decimal shown.

√516 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √516 the pattern is [22; 1, 2, 1, 1, 14, 1, 1, 2, 1, 44] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √516 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 − 516y² = 1. Its smallest solution in positive whole numbers is x = 16,855, y = 742.

√516 in geometry and everyday measurements

• A square garage floor of 516 square feet measures about 22.72 ft (22 ft 9 in) per side, and its corner-to-corner diagonal is √1032 ≈ 32.1 ft.
• 516 is not a sum of two whole-number squares — the prime factor 3 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √516 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 16 × 16 box, because 2² + 16² + 16² = 516.
• Since √516 = 2√129, a length of √516 is exactly 2 copies of the length √129 laid end to end.

Square roots near √516 and related roots

• The cube root of 516 is about 8.020779.
• Because 516 = 4 × 129, the root is twice √129: 2 × 11.357817 ≈ 22.715633.

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