Square Root of 317

√317 at a glance

Exact value

√317

Decimal (10 places)

17.8044938148

Rounded

17.8 · 17.80 · 17.804

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.804494

Prime factorization

317

Cube root

6.818462

How to simplify √317

317 is a prime number, so its only factors are 1 and 317. There is no perfect-square factor to pull out, which means √317 is already in its simplest radical form.

The square root of any prime is irrational. If √317 were a fraction a/b in lowest terms, then a² = 317b², so 317 would divide a — and then 317 would divide b too, contradicting “lowest terms.” That is why the decimal 17.8044938148 is only a rounded value.

Where √317 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √317 lies between 17 and 18. 317 is 28 above 289 and 7 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.8000 (0.03% low)
• Tangent from 17, i.e. 17 + 28 ÷ 34: 17.8235 (0.11% high)
• Tangent from 18, i.e. 18 − 7 ÷ 36: 17.8056 (0.01% high)

For √317 the tangent at 18 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 317 is just 7 below 324.

Finding √317 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 317: following the tangent line down to zero simplifies to averaging x with 317 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √317 = 17.8044938148 to every decimal shown.

√317 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √317 the pattern is [17; 1, 4, 8, 1, 2, 2, 1, 8, 4, 1, 34] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √317 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 − 317y² = 1. Its smallest solution in positive whole numbers is x = 248,678,907,849, y = 13,967,198,980. Because the period is odd, the equation with −1 on the right also has a solution: 352,618² − 317 × 19,805² = −1.

√317 in geometry and everyday measurements

• A square patio or deck of 317 square feet is about 17.8 ft (17 ft 10 in) on each side, so edging all the way around takes 4 × √317 ≈ 71.2 ft.
• 317 = 11² + 14², so by the Pythagorean theorem √317 is the diagonal of a 11 × 14 rectangle — and the distance between the points (0, 0) and (11, 14) on a grid.

Square roots near √317 and related roots

• The cube root of 317 is about 6.818462.
• Squaring undoes the root: (√317)² = 317, while 317² = 100,489 — the number whose square root is 317.

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