Square Root of 205

√205 at a glance

Exact value

√205

Decimal (10 places)

14.3178210633

Rounded

14.3 · 14.32 · 14.318

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.317821

Prime factorization

5 × 41

Cube root

5.896369

How to simplify √205

The prime factorization of 205 is 5 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √205 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 205, 5 and 41 appear an odd number of times, so √205 is irrational and 14.3178210633 is a rounded value.

Where √205 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √205 lies between 14 and 15. 205 is 9 above 196 and 20 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.3103 (0.05% low)
• Tangent from 14, i.e. 14 + 9 ÷ 28: 14.3214 (0.03% high)
• Tangent from 15, i.e. 15 − 20 ÷ 30: 14.3333 (0.11% high)

For √205 the tangent at 14 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 205 is just 9 above 196.

Finding √205 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 205: following the tangent line down to zero simplifies to averaging x with 205 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √205 = 14.3178210633 to every decimal shown.

√205 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √205 the pattern is [14; 3, 6, 1, 4, 1, 6, 3, 28] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √205 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 − 205y² = 1. Its smallest solution in positive whole numbers is x = 39,689, y = 2,772.

√205 in geometry and everyday measurements

• A square patio or deck of 205 square feet is about 14.32 ft (14 ft 4 in) on each side, so edging all the way around takes 4 × √205 ≈ 57.3 ft.
• 205 = 3² + 14² = 6² + 13², so by the Pythagorean theorem √205 is the diagonal of rectangles measuring 3 × 14 and 6 × 13 — and the distance between the points (0, 0) and (3, 14) on a grid.

Square roots near √205 and related roots

• The cube root of 205 is about 5.896369.
• Four times the radicand doubles the root: √820 = 2 × √205 ≈ 28.635642.

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