Square Root of 280

√280 at a glance

Exact value

2√70

Decimal (10 places)

16.7332005307

Rounded

16.7 · 16.73 · 16.733

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.733201

Prime factorization

2³ × 5 × 7

Cube root

6.542133

How to simplify √280

Look for the largest perfect square that divides 280. Here it is 4 (2²), because 280 = 4 × 70 and 70 has no square factor left:

The prime factorization tells the same story: 280 = 2³ × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 5 × 7 stays inside.

Check: (2√70)² = 2² × 70 = 4 × 70 = 280. As a decimal, 2√70 = 2 × 8.3666002653 ≈ 16.7332005307.

Where √280 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √280 lies between 16 and 17. 280 is 24 above 256 and 9 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.7273 (0.04% low)
• Tangent from 16, i.e. 16 + 24 ÷ 32: 16.7500 (0.1% high)
• Tangent from 17, i.e. 17 − 9 ÷ 34: 16.7353 (0.01% high)

For √280 the tangent at 17 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 280 is just 9 below 289.

Finding √280 with the Babylonian method

Picture a rectangle with an area of 280 and one side x; the other side must be 280 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √280.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √280 = 16.7332005307 to every decimal shown.

√280 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √280 the pattern is [16; 1, 2, 1, 2, 1, 32] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √280 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 − 280y² = 1. Its smallest solution in positive whole numbers is x = 251, y = 15.

√280 in geometry and everyday measurements

• A square patio or deck of 280 square feet is about 16.73 ft (16 ft 9 in) on each side, so edging all the way around takes 4 × √280 ≈ 66.9 ft.
• 280 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √280 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 10 × 12 box, because 6² + 10² + 12² = 280.
• Since √280 = 2√70, a length of √280 is exactly 2 copies of the length √70 laid end to end.

Square roots near √280 and related roots

• The cube root of 280 is about 6.542133.
• Because 280 = 4 × 70, the root is twice √70: 2 × 8.3666 ≈ 16.733201.

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