Square Root of 345

√345 at a glance

Exact value

√345

Decimal (10 places)

18.5741756210

Rounded

18.6 · 18.57 · 18.574

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.574176

Prime factorization

3 × 5 × 23

Cube root

7.013579

How to simplify √345

The prime factorization of 345 is 3 × 5 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √345 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 345, 3, 5 and 23 appear an odd number of times, so √345 is irrational and 18.5741756210 is a rounded value.

Where √345 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √345 lies between 18 and 19. 345 is 21 above 324 and 16 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.5676 (0.04% low)
• Tangent from 18, i.e. 18 + 21 ÷ 36: 18.5833 (0.05% high)
• Tangent from 19, i.e. 19 − 16 ÷ 38: 18.5789 (0.03% high)

For √345 the tangent at 19 wins, missing by only 0.0048. Tangent estimates shine when the number sits close to a perfect square — here 345 is just 16 below 361.

Finding √345 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 345: following the tangent line down to zero simplifies to averaging x with 345 ÷ x.

Start from the nearest whole number, 19 (19² = 361):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √345 = 18.5741756210 to every decimal shown.

√345 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √345 the pattern is [18; 1, 1, 2, 1, 6, 1, 2, 1, 1, 36] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √345 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 − 345y² = 1. Its smallest solution in positive whole numbers is x = 6,761, y = 364.

√345 in geometry and everyday measurements

• A square patio or deck of 345 square feet is about 18.57 ft (18 ft 7 in) on each side, so edging all the way around takes 4 × √345 ≈ 74.3 ft.
• 345 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √345 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 8 × 16 box, because 5² + 8² + 16² = 345.

Square roots near √345 and related roots

• The cube root of 345 is about 7.013579.
• Squaring undoes the root: (√345)² = 345, while 345² = 119,025 — the number whose square root is 345.

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