Square Root of 284

√284 at a glance

Exact value

2√71

Decimal (10 places)

16.8522995464

Rounded

16.9 · 16.85 · 16.852

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.852300

Prime factorization

2² × 71

Cube root

6.573138

How to simplify √284

Look for the largest perfect square that divides 284. Here it is 4 (2²), because 284 = 4 × 71 and 71 has no square factor left:

The prime factorization tells the same story: 284 = 2² × 71. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 71 stays inside.

Check: (2√71)² = 2² × 71 = 4 × 71 = 284. As a decimal, 2√71 = 2 × 8.4261497732 ≈ 16.8522995464.

Where √284 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √284 lies between 16 and 17. 284 is 28 above 256 and 5 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.8485 (0.02% low)
• Tangent from 16, i.e. 16 + 28 ÷ 32: 16.8750 (0.13% high)
• Tangent from 17, i.e. 17 − 5 ÷ 34: 16.8529 (0% high)

For √284 the tangent at 17 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 284 is just 5 below 289.

Finding √284 with the Babylonian method

Picture a rectangle with an area of 284 and one side x; the other side must be 284 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √284.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √284 = 16.8522995464 to every decimal shown.

√284 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √284 the pattern is [16; 1, 5, 1, 3, 2, 1, 4, 8, 4, 1, 2, 3, …] 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 √284 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 − 284y² = 1. Its smallest solution in positive whole numbers is x = 24,220,799, y = 1,437,240.

√284 in geometry and everyday measurements

• A square patio or deck of 284 square feet is about 16.85 ft (16 ft 10 in) on each side, so edging all the way around takes 4 × √284 ≈ 67.4 ft.
• 284 is not a sum of two whole-number squares — the prime factor 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √284 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √284 as its space diagonal.
• Since √284 = 2√71, a length of √284 is exactly 2 copies of the length √71 laid end to end.

Square roots near √284 and related roots

• The cube root of 284 is about 6.573138.
• Because 284 = 4 × 71, the root is twice √71: 2 × 8.42615 ≈ 16.8523.

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