Square Root of 337

√337 at a glance

Exact value

√337

Decimal (10 places)

18.3575597507

Rounded

18.4 · 18.36 · 18.358

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.357560

Prime factorization

337

Cube root

6.958943

How to simplify √337

337 is a prime number, so its only factors are 1 and 337. There is no perfect-square factor to pull out, which means √337 is already in its simplest radical form.

The square root of any prime is irrational. If √337 were a fraction a/b in lowest terms, then a² = 337b², so 337 would divide a — and then 337 would divide b too, contradicting “lowest terms.” That is why the decimal 18.3575597507 is only a rounded value.

Where √337 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √337 lies between 18 and 19. 337 is 13 above 324 and 24 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.3514 (0.03% low)
• Tangent from 18, i.e. 18 + 13 ÷ 36: 18.3611 (0.02% high)
• Tangent from 19, i.e. 19 − 24 ÷ 38: 18.3684 (0.06% high)

For √337 the tangent at 18 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 337 is just 13 above 324.

Finding √337 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 337: following the tangent line down to zero simplifies to averaging x with 337 ÷ x.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √337 = 18.3575597507 to every decimal shown.

√337 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √337 the pattern is [18; 2, 1, 3, 1, 11, 2, 4, 1, 3, 3, 1, 4, …] with the block of 19 terms after the semicolon repeating forever (only the first 12 of the 19 are shown). A pattern that never ends is one more proof that √337 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 − 337y² = 1. Its smallest solution in positive whole numbers is x = 2,063,810,353,129,713,793, y = 112,422,913,565,764,752 — 19 digits for x, even though 337 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 1,015,827,336² − 337 × 55,335,641² = −1.

√337 in geometry and everyday measurements

• A square patio or deck of 337 square feet is about 18.36 ft (18 ft 4 in) on each side, so edging all the way around takes 4 × √337 ≈ 73.4 ft.
• 337 = 9² + 16², so by the Pythagorean theorem √337 is the diagonal of a 9 × 16 rectangle — and the distance between the points (0, 0) and (9, 16) on a grid.

Square roots near √337 and related roots

• The cube root of 337 is about 6.958943.
• Squaring undoes the root: (√337)² = 337, while 337² = 113,569 — the number whose square root is 337.

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