Square Root of 279

√279 at a glance

Exact value

3√31

Decimal (10 places)

16.7032930885

Rounded

16.7 · 16.70 · 16.703

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.703293

Prime factorization

3² × 31

Cube root

6.534335

How to simplify √279

Look for the largest perfect square that divides 279. Here it is 9 (3²), because 279 = 9 × 31 and 31 has no square factor left:

The prime factorization tells the same story: 279 = 3² × 31. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 31 stays inside.

Check: (3√31)² = 3² × 31 = 9 × 31 = 279. As a decimal, 3√31 = 3 × 5.5677643628 ≈ 16.7032930885.

Where √279 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √279 lies between 16 and 17. 279 is 23 above 256 and 10 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.6970 (0.04% low)
• Tangent from 16, i.e. 16 + 23 ÷ 32: 16.7188 (0.09% high)
• Tangent from 17, i.e. 17 − 10 ÷ 34: 16.7059 (0.02% high)

For √279 the tangent at 17 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 279 is just 10 below 289.

Finding √279 with the Babylonian method

If a guess is too big, 279 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√279) in one step.

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 √279 = 16.7032930885 to every decimal shown.

√279 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √279 the pattern is [16; 1, 2, 2, 1, 2, 2, 1, 32] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √279 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 − 279y² = 1. Its smallest solution in positive whole numbers is x = 1,520, y = 91.

√279 in geometry and everyday measurements

• A square patio or deck of 279 square feet is about 16.7 ft (16 ft 8 in) on each side, so edging all the way around takes 4 × √279 ≈ 66.8 ft.
• 279 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √279 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 √279 as its space diagonal.
• Since √279 = 3√31, a length of √279 is exactly 3 copies of the length √31 laid end to end.

Square roots near √279 and related roots

• The cube root of 279 is about 6.534335.
• Squaring undoes the root: (√279)² = 279, while 279² = 77,841 — the number whose square root is 279.

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