Square Root of 343

√343 at a glance

Exact value

7√7

Decimal (10 places)

18.5202591775

Rounded

18.5 · 18.52 · 18.520

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.520259

Prime factorization

7³

Cube root

7

How to simplify √343

Look for the largest perfect square that divides 343. Here it is 49 (7²), because 343 = 49 × 7 and 7 has no square factor left:

The prime factorization tells the same story: 343 = 7³. Each pair of equal primes leaves the radical as one factor, so 7 comes out and 7 stays inside.

Check: (7√7)² = 7² × 7 = 49 × 7 = 343. As a decimal, 7√7 = 7 × 2.6457513111 ≈ 18.5202591775.

Where √343 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √343 lies between 18 and 19. 343 is 19 above 324 and 18 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.5135 (0.04% low)
• Tangent from 18, i.e. 18 + 19 ÷ 36: 18.5278 (0.04% high)
• Tangent from 19, i.e. 19 − 18 ÷ 38: 18.5263 (0.03% high)

For √343 the tangent at 19 wins, missing by only 0.0061. Tangent estimates shine when the number sits close to a perfect square — here 343 is just 18 below 361.

Finding √343 with the Babylonian method

If a guess is too big, 343 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√343) in one step.

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 √343 = 18.5202591775 to every decimal shown.

√343 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √343 the pattern is [18; 1, 1, 11, 1, 5, 3, 1, 17, 1, 3, 5, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √343 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 − 343y² = 1. Its smallest solution in positive whole numbers is x = 130,576,328, y = 7,050,459.

√343 in geometry and everyday measurements

• A square patio or deck of 343 square feet is about 18.52 ft (18 ft 6 in) on each side, so edging all the way around takes 4 × √343 ≈ 74.1 ft.
• 343 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √343 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 √343 as its space diagonal.
• Since √343 = 7√7, a length of √343 is exactly 7 copies of the length √7 laid end to end.

Square roots near √343 and related roots

• The cube root of 343 is exactly 7 — 343 is a perfect cube as well (7³).
• Squaring undoes the root: (√343)² = 343, while 343² = 117,649 — the number whose square root is 343.

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