Square Root of 294

√294 at a glance

Exact value

7√6

Decimal (10 places)

17.1464281995

Rounded

17.1 · 17.15 · 17.146

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.146428

Prime factorization

2 × 3 × 7²

Cube root

6.649400

How to simplify √294

Look for the largest perfect square that divides 294. Here it is 49 (7²), because 294 = 49 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 294 = 2 × 3 × 7². Each pair of equal primes leaves the radical as one factor, so 7 comes out and 2 × 3 stays inside.

Check: (7√6)² = 7² × 6 = 49 × 6 = 294. As a decimal, 7√6 = 7 × 2.4494897428 ≈ 17.1464281995.

Where √294 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √294 lies between 17 and 18. 294 is 5 above 289 and 30 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.1429 (0.02% low)
• Tangent from 17, i.e. 17 + 5 ÷ 34: 17.1471 (0% high)
• Tangent from 18, i.e. 18 − 30 ÷ 36: 17.1667 (0.12% high)

For √294 the tangent at 17 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 294 is just 5 above 289.

Finding √294 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 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √294 = 17.1464281995 to every decimal shown.

√294 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √294 the pattern is [17; 6, 1, 4, 1, 6, 34] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √294 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 − 294y² = 1. Its smallest solution in positive whole numbers is x = 4,801, y = 280.

√294 in geometry and everyday measurements

• A square patio or deck of 294 square feet is about 17.15 ft (17 ft 2 in) on each side, so edging all the way around takes 4 × √294 ≈ 68.6 ft.
• 294 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √294 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 17 box, because 1² + 2² + 17² = 294.
• Since √294 = 7√6, a length of √294 is exactly 7 copies of the length √6 laid end to end.

Square roots near √294 and related roots

• The cube root of 294 is about 6.649400.
• Squaring undoes the root: (√294)² = 294, while 294² = 86,436 — the number whose square root is 294.

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