Square Root of 117

√117 at a glance

Exact value

3√13

Decimal (10 places)

10.8166538264

Rounded

10.8 · 10.82 · 10.817

Perfect square?

No — between 10² and 11²

Rational?

Irrational

Both square roots

±10.816654

Prime factorization

3² × 13

Cube root

4.890973

How to simplify √117

Look for the largest perfect square that divides 117. Here it is 9 (3²), because 117 = 9 × 13 and 13 has no square factor left:

The prime factorization tells the same story: 117 = 3² × 13. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 13 stays inside.

Check: (3√13)² = 3² × 13 = 9 × 13 = 117. As a decimal, 3√13 = 3 × 3.6055512755 ≈ 10.8166538264.

Where √117 sits between perfect squares

100 = 10² and 121 = 11² are the nearest perfect squares, so √117 lies between 10 and 11. 117 is 17 above 100 and 4 below 121, so the root is closer to 11.

• Straight line between 100 and 121: 10.8095 (0.07% low)
• Tangent from 10, i.e. 10 + 17 ÷ 20: 10.8500 (0.31% high)
• Tangent from 11, i.e. 11 − 4 ÷ 22: 10.8182 (0.01% high)

For √117 the tangent at 11 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 117 is just 4 below 121.

Finding √117 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 117: following the tangent line down to zero simplifies to averaging x with 117 ÷ x.

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 √117 = 10.8166538264 to every decimal shown.

√117 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √117 the pattern is [10; 1, 4, 2, 4, 1, 20] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √117 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 − 117y² = 1. Its smallest solution in positive whole numbers is x = 649, y = 60.

√117 in geometry and everyday measurements

• A square room or garden bed covering 117 square feet measures about 10.82 ft (10 ft 10 in) along each wall.
• 117 = 6² + 9², so by the Pythagorean theorem √117 is the diagonal of a 6 × 9 rectangle — and the distance between the points (0, 0) and (6, 9) on a grid.
• Since √117 = 3√13, a length of √117 is exactly 3 copies of the length √13 laid end to end.

Square roots near √117 and related roots

• The cube root of 117 is about 4.890973.
• Four times the radicand doubles the root: √468 = 2 × √117 ≈ 21.633308.

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