Square Root of 285

√285 at a glance

Exact value

√285

Decimal (10 places)

16.8819430161

Rounded

16.9 · 16.88 · 16.882

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.881943

Prime factorization

3 × 5 × 19

Cube root

6.580844

How to simplify √285

The prime factorization of 285 is 3 × 5 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √285 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 285, 3, 5 and 19 appear an odd number of times, so √285 is irrational and 16.8819430161 is a rounded value.

Where √285 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √285 lies between 16 and 17. 285 is 29 above 256 and 4 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.8788 (0.02% low)
• Tangent from 16, i.e. 16 + 29 ÷ 32: 16.9063 (0.14% high)
• Tangent from 17, i.e. 17 − 4 ÷ 34: 16.8824 (0% high)

For √285 the tangent at 17 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 285 is just 4 below 289.

Finding √285 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 285: following the tangent line down to zero simplifies to averaging x with 285 ÷ x.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √285 = 16.8819430161 to every decimal shown.

√285 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √285 the pattern is [16; 1, 7, 2, 7, 1, 32] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √285 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 − 285y² = 1. Its smallest solution in positive whole numbers is x = 2,431, y = 144.

√285 in geometry and everyday measurements

• A square patio or deck of 285 square feet is about 16.88 ft (16 ft 11 in) on each side, so edging all the way around takes 4 × √285 ≈ 67.5 ft.
• 285 is not a sum of two whole-number squares — the prime factor 3 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √285 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 16 box, because 2² + 5² + 16² = 285.

Square roots near √285 and related roots

• The cube root of 285 is about 6.580844.
• Squaring undoes the root: (√285)² = 285, while 285² = 81,225 — the number whose square root is 285.

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