Square Root of 313

√313 at a glance

Exact value

√313

Decimal (10 places)

17.6918060130

Rounded

17.7 · 17.69 · 17.692

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.691806

Prime factorization

313

Cube root

6.789661

How to simplify √313

313 is a prime number, so its only factors are 1 and 313. There is no perfect-square factor to pull out, which means √313 is already in its simplest radical form.

The square root of any prime is irrational. If √313 were a fraction a/b in lowest terms, then a² = 313b², so 313 would divide a — and then 313 would divide b too, contradicting “lowest terms.” That is why the decimal 17.6918060130 is only a rounded value.

Where √313 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √313 lies between 17 and 18. 313 is 24 above 289 and 11 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.6857 (0.03% low)
• Tangent from 17, i.e. 17 + 24 ÷ 34: 17.7059 (0.08% high)
• Tangent from 18, i.e. 18 − 11 ÷ 36: 17.6944 (0.01% high)

For √313 the tangent at 18 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 313 is just 11 below 324.

Finding √313 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 313: following the tangent line down to zero simplifies to averaging x with 313 ÷ 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 √313 = 17.6918060130 to every decimal shown.

√313 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √313 the pattern is [17; 1, 2, 4, 11, 1, 1, 3, 2, 2, 3, 1, 1, …] with the block of 17 terms after the semicolon repeating forever (only the first 12 of the 17 are shown). A pattern that never ends is one more proof that √313 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 − 313y² = 1. Its smallest solution in positive whole numbers is x = 32,188,120,829,134,849, y = 1,819,380,158,564,160 — 17 digits for x, even though 313 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: 126,862,368² − 313 × 7,170,685² = −1.

√313 in geometry and everyday measurements

• A square patio or deck of 313 square feet is about 17.69 ft (17 ft 8 in) on each side, so edging all the way around takes 4 × √313 ≈ 70.8 ft.
• 313 = 12² + 13², so by the Pythagorean theorem √313 is the diagonal of a 12 × 13 rectangle — and the distance between the points (0, 0) and (12, 13) on a grid.

Square roots near √313 and related roots

• The cube root of 313 is about 6.789661.
• Squaring undoes the root: (√313)² = 313, while 313² = 97,969 — the number whose square root is 313.

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