Square Root of 265

√265 at a glance

Exact value

√265

Decimal (10 places)

16.2788205961

Rounded

16.3 · 16.28 · 16.279

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.278821

Prime factorization

5 × 53

Cube root

6.423158

How to simplify √265

The prime factorization of 265 is 5 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √265 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 265, 5 and 53 appear an odd number of times, so √265 is irrational and 16.2788205961 is a rounded value.

Where √265 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √265 lies between 16 and 17. 265 is 9 above 256 and 24 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.2727 (0.04% low)
• Tangent from 16, i.e. 16 + 9 ÷ 32: 16.2813 (0.01% high)
• Tangent from 17, i.e. 17 − 24 ÷ 34: 16.2941 (0.09% high)

For √265 the tangent at 16 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 265 is just 9 above 256.

Finding √265 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 265: following the tangent line down to zero simplifies to averaging x with 265 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √265 = 16.2788205961 to every decimal shown.

√265 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √265 the pattern is [16; 3, 1, 1, 2, 2, 1, 1, 3, 32] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √265 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 − 265y² = 1. Its smallest solution in positive whole numbers is x = 73,738,369, y = 4,529,712. Because the period is odd, the equation with −1 on the right also has a solution: 6,072² − 265 × 373² = −1.

√265 in geometry and everyday measurements

• A square patio or deck of 265 square feet is about 16.28 ft (16 ft 3 in) on each side, so edging all the way around takes 4 × √265 ≈ 65.1 ft.
• 265 = 3² + 16² = 11² + 12², so by the Pythagorean theorem √265 is the diagonal of rectangles measuring 3 × 16 and 11 × 12 — and the distance between the points (0, 0) and (3, 16) on a grid.

Square roots near √265 and related roots

• The cube root of 265 is about 6.423158.
• Squaring undoes the root: (√265)² = 265, while 265² = 70,225 — the number whose square root is 265.

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