√246 at a glance
- Exact value
- √246
- Decimal (10 places)
- 15.6843871414
- Rounded
- 15.7 · 15.68 · 15.684
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.684387
- Prime factorization
- 2 × 3 × 41
- Cube root
- 6.265827
How to simplify √246
The prime factorization of 246 is 2 × 3 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √246 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 246, 2, 3 and 41 appear an odd number of times, so √246 is irrational and 15.6843871414 is a rounded value.
Where √246 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √246 lies between 15 and 16. 246 is 21 above 225 and 10 below 256, so the root is closer to 16.
- Straight line between 225 and 256: 15.6774 (0.04% low)
- Tangent from 15, i.e. 15 + 21 ÷ 30: 15.7000 (0.1% high)
- Tangent from 16, i.e. 16 − 10 ÷ 32: 15.6875 (0.02% high)
For √246 the tangent at 16 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 246 is just 10 below 256.
Finding √246 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, 16 (16² = 256):
| Step | Guess x | 246 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 16.0000000000 | 15.3750000000 | 15.6875000000 | 2 |
| 2 | 15.6875000000 | 15.6812749004 | 15.6843874502 | 6 |
| 3 | 15.6843874502 | 15.6843868325 | 15.6843871414 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √246 = 15.6843871414 to every decimal shown.
√246 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √246 the pattern is [15; 1, 2, 5, 1, 14, 1, 5, 2, 1, 30] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √246 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 15/1 | 15.0000000000 | 6.8 × 10⁻¹ |
| 16/1 | 16.0000000000 | 3.2 × 10⁻¹ |
| 47/3 | 15.6666666667 | 1.8 × 10⁻² |
| 251/16 | 15.6875000000 | 3.1 × 10⁻³ |
| 298/19 | 15.6842105263 | 1.8 × 10⁻⁴ |
| 4,423/282 | 15.6843971631 | 1.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 246y² = 1. Its smallest solution in positive whole numbers is x = 88,805, y = 5,662.
√246 in geometry and everyday measurements
- A square patio or deck of 246 square feet is about 15.68 ft (15 ft 8 in) on each side, so edging all the way around takes 4 × √246 ≈ 62.7 ft.
- 246 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 √246 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 14 box, because 1² + 7² + 14² = 246.
Square roots near √246 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √243 | 9√3 | 15.5885 | No |
| √244 | 2√61 | 15.6205 | No |
| √245 | 7√5 | 15.6525 | No |
| √246 | √246 | 15.6844 | No |
| √247 | √247 | 15.7162 | No |
| √248 | 2√62 | 15.7480 | No |
| √249 | √249 | 15.7797 | No |
- The cube root of 246 is about 6.265827.
- Four times the radicand doubles the root: √984 = 2 × √246 ≈ 31.368774.
Frequently asked questions
What is the square root of 246?
The square root of 246 is √246, about 15.6843871414. The negative root, −15.684387, also squares to 246.
Is the square root of 246 rational or irrational?
Irrational. 246 is not a perfect square — it falls between 225 and 256 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √246 be simplified?
No. 246 = 2 × 3 × 41 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √246 rounded to two decimal places?
√246 ≈ 15.68 to two decimal places (15.7 to one, 15.684 to three). Check: 15.68² = 245.8624, close to 246.