Square Root of 249

√249 at a glance

Exact value

√249

Decimal (10 places)

15.7797338381

Rounded

15.8 · 15.78 · 15.780

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.779734

Prime factorization

3 × 83

Cube root

6.291195

How to simplify √249

The prime factorization of 249 is 3 × 83. Every prime appears only once, so there is no pair to bring outside the radical — √249 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 249, 3 and 83 appear an odd number of times, so √249 is irrational and 15.7797338381 is a rounded value.

Where √249 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √249 lies between 15 and 16. 249 is 24 above 225 and 7 below 256, so the root is closer to 16.

• Straight line between 225 and 256: 15.7742 (0.04% low)
• Tangent from 15, i.e. 15 + 24 ÷ 30: 15.8000 (0.13% high)
• Tangent from 16, i.e. 16 − 7 ÷ 32: 15.7813 (0.01% high)

For √249 the tangent at 16 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 249 is just 7 below 256.

Finding √249 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 249: following the tangent line down to zero simplifies to averaging x with 249 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √249 = 15.7797338381 to every decimal shown.

√249 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √249 the pattern is [15; 1, 3, 1, 1, 5, 1, 3, 10, 3, 1, 5, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √249 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 − 249y² = 1. Its smallest solution in positive whole numbers is x = 8,553,815, y = 542,076.

√249 in geometry and everyday measurements

• A square patio or deck of 249 square feet is about 15.78 ft (15 ft 9 in) on each side, so edging all the way around takes 4 × √249 ≈ 63.1 ft.
• 249 is not a sum of two whole-number squares — the prime factor 3 and 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √249 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 14 box, because 2² + 7² + 14² = 249.

Square roots near √249 and related roots

• The cube root of 249 is about 6.291195.
• Four times the radicand doubles the root: √996 = 2 × √249 ≈ 31.559468.

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