Square Root of 316

√316 at a glance

Exact value

2√79

Decimal (10 places)

17.7763888346

Rounded

17.8 · 17.78 · 17.776

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.776389

Prime factorization

2² × 79

Cube root

6.811285

How to simplify √316

Look for the largest perfect square that divides 316. Here it is 4 (2²), because 316 = 4 × 79 and 79 has no square factor left:

The prime factorization tells the same story: 316 = 2² × 79. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 79 stays inside.

Check: (2√79)² = 2² × 79 = 4 × 79 = 316. As a decimal, 2√79 = 2 × 8.8881944173 ≈ 17.7763888346.

Where √316 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √316 lies between 17 and 18. 316 is 27 above 289 and 8 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.7714 (0.03% low)
• Tangent from 17, i.e. 17 + 27 ÷ 34: 17.7941 (0.1% high)
• Tangent from 18, i.e. 18 − 8 ÷ 36: 17.7778 (0.01% high)

For √316 the tangent at 18 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 316 is just 8 below 324.

Finding √316 with the Babylonian method

Picture a rectangle with an area of 316 and one side x; the other side must be 316 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √316.

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 √316 = 17.7763888346 to every decimal shown.

√316 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √316 the pattern is [17; 1, 3, 2, 8, 2, 3, 1, 34] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √316 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 − 316y² = 1. Its smallest solution in positive whole numbers is x = 12,799, y = 720.

√316 in geometry and everyday measurements

• A square patio or deck of 316 square feet is about 17.78 ft (17 ft 9 in) on each side, so edging all the way around takes 4 × √316 ≈ 71.1 ft.
• 316 is not a sum of two whole-number squares — the prime factor 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √316 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √316 as its space diagonal.
• Since √316 = 2√79, a length of √316 is exactly 2 copies of the length √79 laid end to end.

Square roots near √316 and related roots

• The cube root of 316 is about 6.811285.
• Because 316 = 4 × 79, the root is twice √79: 2 × 8.888194 ≈ 17.776389.

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