Square Root of 348

√348 at a glance

Exact value

2√87

Decimal (10 places)

18.6547581062

Rounded

18.7 · 18.65 · 18.655

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.654758

Prime factorization

2² × 3 × 29

Cube root

7.033850

How to simplify √348

Look for the largest perfect square that divides 348. Here it is 4 (2²), because 348 = 4 × 87 and 87 has no square factor left:

The prime factorization tells the same story: 348 = 2² × 3 × 29. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 29 stays inside.

Check: (2√87)² = 2² × 87 = 4 × 87 = 348. As a decimal, 2√87 = 2 × 9.3273790531 ≈ 18.6547581062.

Where √348 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √348 lies between 18 and 19. 348 is 24 above 324 and 13 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.6486 (0.03% low)
• Tangent from 18, i.e. 18 + 24 ÷ 36: 18.6667 (0.06% high)
• Tangent from 19, i.e. 19 − 13 ÷ 38: 18.6579 (0.02% high)

For √348 the tangent at 19 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 348 is just 13 below 361.

Finding √348 with the Babylonian method

Picture a rectangle with an area of 348 and one side x; the other side must be 348 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √348.

Start from the nearest whole number, 19 (19² = 361):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √348 = 18.6547581062 to every decimal shown.

√348 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √348 the pattern is [18; 1, 1, 1, 8, 1, 1, 1, 36] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √348 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 − 348y² = 1. Its smallest solution in positive whole numbers is x = 1,567, y = 84.

√348 in geometry and everyday measurements

• A square patio or deck of 348 square feet is about 18.65 ft (18 ft 8 in) on each side, so edging all the way around takes 4 × √348 ≈ 74.6 ft.
• 348 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 √348 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 √348 as its space diagonal.
• Since √348 = 2√87, a length of √348 is exactly 2 copies of the length √87 laid end to end.

Square roots near √348 and related roots

• The cube root of 348 is about 7.033850.
• Because 348 = 4 × 87, the root is twice √87: 2 × 9.327379 ≈ 18.654758.

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