Square Root of 181

√181 at a glance

Exact value

√181

Decimal (10 places)

13.4536240471

Rounded

13.5 · 13.45 · 13.454

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.453624

Prime factorization

181

Cube root

5.656653

How to simplify √181

181 is a prime number, so its only factors are 1 and 181. There is no perfect-square factor to pull out, which means √181 is already in its simplest radical form.

The square root of any prime is irrational. If √181 were a fraction a/b in lowest terms, then a² = 181b², so 181 would divide a — and then 181 would divide b too, contradicting “lowest terms.” That is why the decimal 13.4536240471 is only a rounded value.

Where √181 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √181 lies between 13 and 14. 181 is 12 above 169 and 15 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.4444 (0.07% low)
• Tangent from 13, i.e. 13 + 12 ÷ 26: 13.4615 (0.06% high)
• Tangent from 14, i.e. 14 − 15 ÷ 28: 13.4643 (0.08% high)

For √181 the tangent at 13 wins, missing by only 0.0079. Tangent estimates shine when the number sits close to a perfect square — here 181 is just 12 above 169.

Finding √181 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 181: following the tangent line down to zero simplifies to averaging x with 181 ÷ x.

Start from the nearest whole number, 13 (13² = 169):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √181 = 13.4536240471 to every decimal shown.

√181 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √181 the pattern is [13; 2, 4, 1, 8, 6, 1, 1, 1, 1, 2, 2, 1, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √181 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 − 181y² = 1. Its smallest solution in positive whole numbers is x = 2,469,645,423,824,185,801, y = 183,567,298,683,461,940 — 19 digits for x, even though 181 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 1,111,225,770² − 181 × 82,596,761² = −1.

√181 in geometry and everyday measurements

• A square room or garden bed covering 181 square feet measures about 13.45 ft (13 ft 5 in) along each wall.
• 181 = 9² + 10², so by the Pythagorean theorem √181 is the diagonal of a 9 × 10 rectangle — and the distance between the points (0, 0) and (9, 10) on a grid.

Square roots near √181 and related roots

• The cube root of 181 is about 5.656653.
• Four times the radicand doubles the root: √724 = 2 × √181 ≈ 26.907248.

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