Square Root of 292

√292 at a glance

Exact value

2√73

Decimal (10 places)

17.0880074906

Rounded

17.1 · 17.09 · 17.088

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.088007

Prime factorization

2² × 73

Cube root

6.634287

How to simplify √292

Look for the largest perfect square that divides 292. Here it is 4 (2²), because 292 = 4 × 73 and 73 has no square factor left:

The prime factorization tells the same story: 292 = 2² × 73. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 73 stays inside.

Check: (2√73)² = 2² × 73 = 4 × 73 = 292. As a decimal, 2√73 = 2 × 8.5440037453 ≈ 17.0880074906.

Where √292 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √292 lies between 17 and 18. 292 is 3 above 289 and 32 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.0857 (0.01% low)
• Tangent from 17, i.e. 17 + 3 ÷ 34: 17.0882 (0% high)
• Tangent from 18, i.e. 18 − 32 ÷ 36: 17.1111 (0.14% high)

For √292 the tangent at 17 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 292 is just 3 above 289.

Finding √292 with the Babylonian method

Picture a rectangle with an area of 292 and one side x; the other side must be 292 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √292.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √292 = 17.0880074906 to every decimal shown.

√292 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √292 the pattern is [17; 11, 2, 1, 3, 8, 3, 1, 2, 11, 34] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √292 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 − 292y² = 1. Its smallest solution in positive whole numbers is x = 2,281,249, y = 133,500.

√292 in geometry and everyday measurements

• A square patio or deck of 292 square feet is about 17.09 ft (17 ft 1 in) on each side, so edging all the way around takes 4 × √292 ≈ 68.4 ft.
• 292 = 6² + 16², so by the Pythagorean theorem √292 is the diagonal of a 6 × 16 rectangle — and the distance between the points (0, 0) and (6, 16) on a grid.
• Since √292 = 2√73, a length of √292 is exactly 2 copies of the length √73 laid end to end.

Square roots near √292 and related roots

• The cube root of 292 is about 6.634287.
• Because 292 = 4 × 73, the root is twice √73: 2 × 8.544004 ≈ 17.088007.

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