Square Root of 261

√261 at a glance

Exact value

3√29

Decimal (10 places)

16.1554944214

Rounded

16.2 · 16.16 · 16.155

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.155494

Prime factorization

3² × 29

Cube root

6.390677

How to simplify √261

Look for the largest perfect square that divides 261. Here it is 9 (3²), because 261 = 9 × 29 and 29 has no square factor left:

The prime factorization tells the same story: 261 = 3² × 29. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 29 stays inside.

Check: (3√29)² = 3² × 29 = 9 × 29 = 261. As a decimal, 3√29 = 3 × 5.3851648071 ≈ 16.1554944214.

Where √261 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √261 lies between 16 and 17. 261 is 5 above 256 and 28 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.1515 (0.02% low)
• Tangent from 16, i.e. 16 + 5 ÷ 32: 16.1563 (0% high)
• Tangent from 17, i.e. 17 − 28 ÷ 34: 16.1765 (0.13% high)

For √261 the tangent at 16 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 261 is just 5 above 256.

Finding √261 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 261: following the tangent line down to zero simplifies to averaging x with 261 ÷ x.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √261 = 16.1554944214 to every decimal shown.

√261 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √261 the pattern is [16; 6, 2, 3, 7, 1, 3, 1, 2, 1, 3, 1, 7, …] 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 √261 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 − 261y² = 1. Its smallest solution in positive whole numbers is x = 192,119,201, y = 11,891,880.

√261 in geometry and everyday measurements

• A square patio or deck of 261 square feet is about 16.16 ft (16 ft 2 in) on each side, so edging all the way around takes 4 × √261 ≈ 64.6 ft.
• 261 = 6² + 15², so by the Pythagorean theorem √261 is the diagonal of a 6 × 15 rectangle — and the distance between the points (0, 0) and (6, 15) on a grid.
• Since √261 = 3√29, a length of √261 is exactly 3 copies of the length √29 laid end to end.

Square roots near √261 and related roots

• The cube root of 261 is about 6.390677.
• Squaring undoes the root: (√261)² = 261, while 261² = 68,121 — the number whose square root is 261.

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