Square Root of 312

√312 at a glance

Exact value

2√78

Decimal (10 places)

17.6635217327

Rounded

17.7 · 17.66 · 17.664

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.663522

Prime factorization

2³ × 3 × 13

Cube root

6.782423

How to simplify √312

Look for the largest perfect square that divides 312. Here it is 4 (2²), because 312 = 4 × 78 and 78 has no square factor left:

The prime factorization tells the same story: 312 = 2³ × 3 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 13 stays inside.

Check: (2√78)² = 2² × 78 = 4 × 78 = 312. As a decimal, 2√78 = 2 × 8.8317608663 ≈ 17.6635217327.

Where √312 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √312 lies between 17 and 18. 312 is 23 above 289 and 12 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.6571 (0.04% low)
• Tangent from 17, i.e. 17 + 23 ÷ 34: 17.6765 (0.07% high)
• Tangent from 18, i.e. 18 − 12 ÷ 36: 17.6667 (0.02% high)

For √312 the tangent at 18 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 312 is just 12 below 324.

Finding √312 with the Babylonian method

Picture a rectangle with an area of 312 and one side x; the other side must be 312 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √312.

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 √312 = 17.6635217327 to every decimal shown.

√312 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √312 the pattern is [17; 1, 1, 1, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √312 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 − 312y² = 1. Its smallest solution in positive whole numbers is x = 53, y = 3.

√312 in geometry and everyday measurements

• A square patio or deck of 312 square feet is about 17.66 ft (17 ft 8 in) on each side, so edging all the way around takes 4 × √312 ≈ 70.7 ft.
• 312 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √312 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 10 × 14 box, because 4² + 10² + 14² = 312.
• Since √312 = 2√78, a length of √312 is exactly 2 copies of the length √78 laid end to end.

Square roots near √312 and related roots

• The cube root of 312 is about 6.782423.
• Because 312 = 4 × 78, the root is twice √78: 2 × 8.831761 ≈ 17.663522.

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