Square Root of 347

√347 at a glance

Exact value

√347

Decimal (10 places)

18.6279360102

Rounded

18.6 · 18.63 · 18.628

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.627936

Prime factorization

347

Cube root

7.027106

How to simplify √347

347 is a prime number, so its only factors are 1 and 347. There is no perfect-square factor to pull out, which means √347 is already in its simplest radical form.

The square root of any prime is irrational. If √347 were a fraction a/b in lowest terms, then a² = 347b², so 347 would divide a — and then 347 would divide b too, contradicting “lowest terms.” That is why the decimal 18.6279360102 is only a rounded value.

Where √347 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √347 lies between 18 and 19. 347 is 23 above 324 and 14 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.6216 (0.03% low)
• Tangent from 18, i.e. 18 + 23 ÷ 36: 18.6389 (0.06% high)
• Tangent from 19, i.e. 19 − 14 ÷ 38: 18.6316 (0.02% high)

For √347 the tangent at 19 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 347 is just 14 below 361.

Finding √347 with the Babylonian method

If a guess is too big, 347 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√347) in one step.

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 √347 = 18.6279360102 to every decimal shown.

√347 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √347 the pattern is [18; 1, 1, 1, 2, 4, 1, 17, 1, 4, 2, 1, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √347 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 − 347y² = 1. Its smallest solution in positive whole numbers is x = 641,602, y = 34,443.

√347 in geometry and everyday measurements

• A square patio or deck of 347 square feet is about 18.63 ft (18 ft 8 in) on each side, so edging all the way around takes 4 × √347 ≈ 74.5 ft.
• 347 is not a sum of two whole-number squares — 347 is itself a prime that is one less than a multiple of 4, which rules that out — so √347 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 15 box, because 1² + 11² + 15² = 347.

Square roots near √347 and related roots

• The cube root of 347 is about 7.027106.
• Squaring undoes the root: (√347)² = 347, while 347² = 120,409 — the number whose square root is 347.

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