Square Root of 67

√67 at a glance

Exact value

√67

Decimal (10 places)

8.1853527719

Rounded

8.2 · 8.19 · 8.185

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.185353

Prime factorization

67

Cube root

4.061548

How to simplify √67

67 is a prime number, so its only factors are 1 and 67. There is no perfect-square factor to pull out, which means √67 is already in its simplest radical form.

The square root of any prime is irrational. If √67 were a fraction a/b in lowest terms, then a² = 67b², so 67 would divide a — and then 67 would divide b too, contradicting “lowest terms.” That is why the decimal 8.1853527719 is only a rounded value.

Where √67 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √67 lies between 8 and 9. 67 is 3 above 64 and 14 below 81, so the root is closer to 8.

• Straight line between 64 and 81: 8.1765 (0.11% low)
• Tangent from 8, i.e. 8 + 3 ÷ 16: 8.1875 (0.03% high)
• Tangent from 9, i.e. 9 − 14 ÷ 18: 8.2222 (0.45% high)

For √67 the tangent at 8 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 67 is just 3 above 64.

Finding √67 with the Babylonian method

If a guess is too big, 67 ÷ 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√67) in one step.

Start from the nearest whole number, 8 (8² = 64):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √67 = 8.1853527719 to every decimal shown.

√67 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √67 the pattern is [8; 5, 2, 1, 1, 7, 1, 1, 2, 5, 16] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √67 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 − 67y² = 1. Its smallest solution in positive whole numbers is x = 48,842, y = 5,967.

√67 in geometry and everyday measurements

• A square room or garden bed covering 67 square feet measures about 8.19 ft (8 ft 2 in) along each wall.
• 67 is not a sum of two whole-number squares — 67 is itself a prime that is one less than a multiple of 4, which rules that out — so √67 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 3 × 7 box, because 3² + 3² + 7² = 67.

Square roots near √67 and related roots

• The cube root of 67 is about 4.061548.
• Four times the radicand doubles the root: √268 = 2 × √67 ≈ 16.370706.

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