Square Root of 272

√272 at a glance

Exact value

4√17

Decimal (10 places)

16.4924225025

Rounded

16.5 · 16.49 · 16.492

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.492423

Prime factorization

2⁴ × 17

Cube root

6.479224

How to simplify √272

Look for the largest perfect square that divides 272. Here it is 16 (4²), because 272 = 16 × 17 and 17 has no square factor left:

The prime factorization tells the same story: 272 = 2⁴ × 17. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 17 stays inside.

272 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √272 = 2√68, and √68 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√17)² = 4² × 17 = 16 × 17 = 272. As a decimal, 4√17 = 4 × 4.1231056256 ≈ 16.4924225025.

Where √272 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √272 lies between 16 and 17. 272 is 16 above 256 and 17 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.4848 (0.05% low)
• Tangent from 16, i.e. 16 + 16 ÷ 32: 16.5000 (0.05% high)
• Tangent from 17, i.e. 17 − 17 ÷ 34: 16.5000 (0.05% high)

For √272 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √272 with the Babylonian method

Picture a rectangle with an area of 272 and one side x; the other side must be 272 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √272.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √272 = 16.4924225025 to every decimal shown.

√272 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √272 the pattern is [16; 2, 32] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √272 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 − 272y² = 1. Its smallest solution in positive whole numbers is x = 33, y = 2.

√272 in geometry and everyday measurements

• A square patio or deck of 272 square feet is about 16.49 ft (16 ft 6 in) on each side, so edging all the way around takes 4 × √272 ≈ 66 ft.
• 272 = 4² + 16², so by the Pythagorean theorem √272 is the diagonal of a 4 × 16 rectangle — and the distance between the points (0, 0) and (4, 16) on a grid.
• Since √272 = 4√17, a length of √272 is exactly 4 copies of the length √17 laid end to end.

Square roots near √272 and related roots

• The cube root of 272 is about 6.479224.
• Because 272 = 4 × 68, the root is twice √68: 2 × 8.246211 ≈ 16.492423.

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