Square Root of 43

√43 at a glance

Exact value

√43

Decimal (10 places)

6.5574385243

Rounded

6.6 · 6.56 · 6.557

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.557439

Prime factorization

43

Cube root

3.503398

How to simplify √43

43 is a prime number, so its only factors are 1 and 43. There is no perfect-square factor to pull out, which means √43 is already in its simplest radical form.

The square root of any prime is irrational. If √43 were a fraction a/b in lowest terms, then a² = 43b², so 43 would divide a — and then 43 would divide b too, contradicting “lowest terms.” That is why the decimal 6.5574385243 is only a rounded value.

Where √43 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √43 lies between 6 and 7. 43 is 7 above 36 and 6 below 49, so the root is closer to 7.

• Straight line between 36 and 49: 6.5385 (0.29% low)
• Tangent from 6, i.e. 6 + 7 ÷ 12: 6.5833 (0.39% high)
• Tangent from 7, i.e. 7 − 6 ÷ 14: 6.5714 (0.21% high)

For √43 the tangent at 7 wins, missing by only 0.014. Tangent estimates shine when the number sits close to a perfect square — here 43 is just 6 below 49.

Finding √43 with the Babylonian method

If a guess is too big, 43 ÷ 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√43) in one step.

Start from the nearest whole number, 7 (7² = 49):

The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √43 = 6.5574385243 to every decimal shown.

√43 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √43 the pattern is [6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √43 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 − 43y² = 1. Its smallest solution in positive whole numbers is x = 3,482, y = 531.

√43 in geometry and everyday measurements

• A square room or garden bed covering 43 square feet measures about 6.56 ft (6 ft 7 in) along each wall.
• 43 is not a sum of two whole-number squares — 43 is itself a prime that is one less than a multiple of 4, which rules that out — so √43 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 5 box, because 3² + 3² + 5² = 43.

Square roots near √43 and related roots

• The cube root of 43 is about 3.503398.
• Four times the radicand doubles the root: √172 = 2 × √43 ≈ 13.114877.

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