Square Root of 185

√185 at a glance

Exact value

√185

Decimal (10 places)

13.6014705087

Rounded

13.6 · 13.60 · 13.601

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.601471

Prime factorization

5 × 37

Cube root

5.698019

How to simplify √185

The prime factorization of 185 is 5 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √185 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 185, 5 and 37 appear an odd number of times, so √185 is irrational and 13.6014705087 is a rounded value.

Where √185 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √185 lies between 13 and 14. 185 is 16 above 169 and 11 below 196, so the root is closer to 14.

• Straight line between 169 and 196: 13.5926 (0.07% low)
• Tangent from 13, i.e. 13 + 16 ÷ 26: 13.6154 (0.1% high)
• Tangent from 14, i.e. 14 − 11 ÷ 28: 13.6071 (0.04% high)

For √185 the tangent at 14 wins, missing by only 0.0057. Tangent estimates shine when the number sits close to a perfect square — here 185 is just 11 below 196.

Finding √185 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 185: following the tangent line down to zero simplifies to averaging x with 185 ÷ x.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √185 = 13.6014705087 to every decimal shown.

√185 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √185 the pattern is [13; 1, 1, 1, 1, 26] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √185 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 − 185y² = 1. Its smallest solution in positive whole numbers is x = 9,249, y = 680. Because the period is odd, the equation with −1 on the right also has a solution: 68² − 185 × 5² = −1.

√185 in geometry and everyday measurements

• A square room or garden bed covering 185 square feet measures about 13.6 ft (13 ft 7 in) along each wall.
• 185 = 4² + 13² = 8² + 11², so by the Pythagorean theorem √185 is the diagonal of rectangles measuring 4 × 13 and 8 × 11 — and the distance between the points (0, 0) and (4, 13) on a grid.

Square roots near √185 and related roots

• The cube root of 185 is about 5.698019.
• Four times the radicand doubles the root: √740 = 2 × √185 ≈ 27.202941.

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