Square Root of 274

√274 at a glance

Exact value

√274

Decimal (10 places)

16.5529453572

Rounded

16.6 · 16.55 · 16.553

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.552945

Prime factorization

2 × 137

Cube root

6.495065

How to simplify √274

The prime factorization of 274 is 2 × 137. Every prime appears only once, so there is no pair to bring outside the radical — √274 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 274, 2 and 137 appear an odd number of times, so √274 is irrational and 16.5529453572 is a rounded value.

Where √274 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √274 lies between 16 and 17. 274 is 18 above 256 and 15 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.5455 (0.05% low)
• Tangent from 16, i.e. 16 + 18 ÷ 32: 16.5625 (0.06% high)
• Tangent from 17, i.e. 17 − 15 ÷ 34: 16.5588 (0.04% high)

For √274 the tangent at 17 wins, missing by only 0.0059. Tangent estimates shine when the number sits close to a perfect square — here 274 is just 15 below 289.

Finding √274 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 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √274 = 16.5529453572 to every decimal shown.

√274 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √274 the pattern is [16; 1, 1, 4, 4, 1, 1, 32] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √274 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 − 274y² = 1. Its smallest solution in positive whole numbers is x = 3,959,299, y = 239,190. Because the period is odd, the equation with −1 on the right also has a solution: 1,407² − 274 × 85² = −1.

√274 in geometry and everyday measurements

• A square patio or deck of 274 square feet is about 16.55 ft (16 ft 7 in) on each side, so edging all the way around takes 4 × √274 ≈ 66.2 ft.
• 274 = 7² + 15², so by the Pythagorean theorem √274 is the diagonal of a 7 × 15 rectangle — and the distance between the points (0, 0) and (7, 15) on a grid.

Square roots near √274 and related roots

• The cube root of 274 is about 6.495065.
• Squaring undoes the root: (√274)² = 274, while 274² = 75,076 — the number whose square root is 274.

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