Square Root of 282

√282 at a glance

Exact value

√282

Decimal (10 places)

16.7928556237

Rounded

16.8 · 16.79 · 16.793

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.792856

Prime factorization

2 × 3 × 47

Cube root

6.557672

How to simplify √282

The prime factorization of 282 is 2 × 3 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √282 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 282, 2, 3 and 47 appear an odd number of times, so √282 is irrational and 16.7928556237 is a rounded value.

Where √282 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √282 lies between 16 and 17. 282 is 26 above 256 and 7 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.7879 (0.03% low)
• Tangent from 16, i.e. 16 + 26 ÷ 32: 16.8125 (0.12% high)
• Tangent from 17, i.e. 17 − 7 ÷ 34: 16.7941 (0.01% high)

For √282 the tangent at 17 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 282 is just 7 below 289.

Finding √282 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √282 = 16.7928556237 to every decimal shown.

√282 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √282 the pattern is [16; 1, 3, 1, 4, 1, 3, 1, 32] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √282 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 − 282y² = 1. Its smallest solution in positive whole numbers is x = 2,351, y = 140.

√282 in geometry and everyday measurements

• A square patio or deck of 282 square feet is about 16.79 ft (16 ft 10 in) on each side, so edging all the way around takes 4 × √282 ≈ 67.2 ft.
• 282 is not a sum of two whole-number squares — the prime factor 3 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √282 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 16 box, because 1² + 5² + 16² = 282.

Square roots near √282 and related roots

• The cube root of 282 is about 6.557672.
• Squaring undoes the root: (√282)² = 282, while 282² = 79,524 — the number whose square root is 282.

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