Square Root of 116

√116 at a glance

Exact value

2√29

Decimal (10 places)

10.7703296143

Rounded

10.8 · 10.77 · 10.770

Perfect square?

No — between 10² and 11²

Rational?

Irrational

Both square roots

±10.770330

Prime factorization

2² × 29

Cube root

4.876999

How to simplify √116

Look for the largest perfect square that divides 116. Here it is 4 (2²), because 116 = 4 × 29 and 29 has no square factor left:

The prime factorization tells the same story: 116 = 2² × 29. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 29 stays inside.

Check: (2√29)² = 2² × 29 = 4 × 29 = 116. As a decimal, 2√29 = 2 × 5.3851648071 ≈ 10.7703296143.

Where √116 sits between perfect squares

100 = 10² and 121 = 11² are the nearest perfect squares, so √116 lies between 10 and 11. 116 is 16 above 100 and 5 below 121, so the root is closer to 11.

• Straight line between 100 and 121: 10.7619 (0.08% low)
• Tangent from 10, i.e. 10 + 16 ÷ 20: 10.8000 (0.28% high)
• Tangent from 11, i.e. 11 − 5 ÷ 22: 10.7727 (0.02% high)

For √116 the tangent at 11 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 116 is just 5 below 121.

Finding √116 with the Babylonian method

Picture a rectangle with an area of 116 and one side x; the other side must be 116 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √116.

Start from the nearest whole number, 11 (11² = 121):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √116 = 10.7703296143 to every decimal shown.

√116 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √116 the pattern is [10; 1, 3, 2, 1, 4, 1, 2, 3, 1, 20] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √116 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 − 116y² = 1. Its smallest solution in positive whole numbers is x = 9,801, y = 910.

√116 in geometry and everyday measurements

• A square room or garden bed covering 116 square feet measures about 10.77 ft (10 ft 9 in) along each wall.
• 116 = 4² + 10², so by the Pythagorean theorem √116 is the diagonal of a 4 × 10 rectangle — and the distance between the points (0, 0) and (4, 10) on a grid.
• Since √116 = 2√29, a length of √116 is exactly 2 copies of the length √29 laid end to end.

Square roots near √116 and related roots

• The cube root of 116 is about 4.876999.
• Four times the radicand doubles the root: √464 = 2 × √116 ≈ 21.540659.

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