Square Root of 260

√260 at a glance

Exact value

2√65

Decimal (10 places)

16.1245154966

Rounded

16.1 · 16.12 · 16.125

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.124515

Prime factorization

2² × 5 × 13

Cube root

6.382504

How to simplify √260

Look for the largest perfect square that divides 260. Here it is 4 (2²), because 260 = 4 × 65 and 65 has no square factor left:

The prime factorization tells the same story: 260 = 2² × 5 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 13 stays inside.

Check: (2√65)² = 2² × 65 = 4 × 65 = 260. As a decimal, 2√65 = 2 × 8.0622577483 ≈ 16.1245154966.

Where √260 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √260 lies between 16 and 17. 260 is 4 above 256 and 29 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.1212 (0.02% low)
• Tangent from 16, i.e. 16 + 4 ÷ 32: 16.1250 (0% high)
• Tangent from 17, i.e. 17 − 29 ÷ 34: 16.1471 (0.14% high)

For √260 the tangent at 16 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 260 is just 4 above 256.

Finding √260 with the Babylonian method

Picture a rectangle with an area of 260 and one side x; the other side must be 260 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √260.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √260 = 16.1245154966 to every decimal shown.

√260 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √260 the pattern is [16; 8, 32] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √260 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 − 260y² = 1. Its smallest solution in positive whole numbers is x = 129, y = 8.

√260 in geometry and everyday measurements

• A square patio or deck of 260 square feet is about 16.12 ft (16 ft 1 in) on each side, so edging all the way around takes 4 × √260 ≈ 64.5 ft.
• 260 = 2² + 16² = 8² + 14², so by the Pythagorean theorem √260 is the diagonal of rectangles measuring 2 × 16 and 8 × 14 — and the distance between the points (0, 0) and (2, 16) on a grid.
• Since √260 = 2√65, a length of √260 is exactly 2 copies of the length √65 laid end to end.

Square roots near √260 and related roots

• The cube root of 260 is about 6.382504.
• Because 260 = 4 × 65, the root is twice √65: 2 × 8.062258 ≈ 16.124515.

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