Square Root of 340

√340 at a glance

Exact value

2√85

Decimal (10 places)

18.4390889146

Rounded

18.4 · 18.44 · 18.439

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.439089

Prime factorization

2² × 5 × 17

Cube root

6.979532

How to simplify √340

Look for the largest perfect square that divides 340. Here it is 4 (2²), because 340 = 4 × 85 and 85 has no square factor left:

The prime factorization tells the same story: 340 = 2² × 5 × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 17 stays inside.

Check: (2√85)² = 2² × 85 = 4 × 85 = 340. As a decimal, 2√85 = 2 × 9.2195444573 ≈ 18.4390889146.

Where √340 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √340 lies between 18 and 19. 340 is 16 above 324 and 21 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.4324 (0.04% low)
• Tangent from 18, i.e. 18 + 16 ÷ 36: 18.4444 (0.03% high)
• Tangent from 19, i.e. 19 − 21 ÷ 38: 18.4474 (0.04% high)

For √340 the tangent at 18 wins, missing by only 0.0054. Tangent estimates shine when the number sits close to a perfect square — here 340 is just 16 above 324.

Finding √340 with the Babylonian method

Picture a rectangle with an area of 340 and one side x; the other side must be 340 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √340.

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 √340 = 18.4390889146 to every decimal shown.

√340 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √340 the pattern is [18; 2, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 3, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √340 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 − 340y² = 1. Its smallest solution in positive whole numbers is x = 285,769, y = 15,498.

√340 in geometry and everyday measurements

• A square patio or deck of 340 square feet is about 18.44 ft (18 ft 5 in) on each side, so edging all the way around takes 4 × √340 ≈ 73.8 ft.
• 340 = 4² + 18² = 12² + 14², so by the Pythagorean theorem √340 is the diagonal of rectangles measuring 4 × 18 and 12 × 14 — and the distance between the points (0, 0) and (4, 18) on a grid.
• Since √340 = 2√85, a length of √340 is exactly 2 copies of the length √85 laid end to end.

Square roots near √340 and related roots

• The cube root of 340 is about 6.979532.
• Because 340 = 4 × 85, the root is twice √85: 2 × 9.219544 ≈ 18.439089.

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