√242 at a glance
- Exact value
- 11√2
- Decimal (10 places)
- 15.5563491861
- Rounded
- 15.6 · 15.56 · 15.556
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.556349
- Prime factorization
- 2 × 11²
- Cube root
- 6.231680
How to simplify √242
Look for the largest perfect square that divides 242. Here it is 121 (11²), because 242 = 121 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 242 = 2 × 11². Each pair of equal primes leaves the radical as one factor, so 11 comes out and 2 stays inside.
Check: (11√2)² = 11² × 2 = 121 × 2 = 242. As a decimal, 11√2 = 11 × 1.4142135624 ≈ 15.5563491861.
Where √242 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √242 lies between 15 and 16. 242 is 17 above 225 and 14 below 256, so the root is closer to 16.
- Straight line between 225 and 256: 15.5484 (0.05% low)
- Tangent from 15, i.e. 15 + 17 ÷ 30: 15.5667 (0.07% high)
- Tangent from 16, i.e. 16 − 14 ÷ 32: 15.5625 (0.04% high)
For √242 the tangent at 16 wins, missing by only 0.0062. Tangent estimates shine when the number sits close to a perfect square — here 242 is just 14 below 256.
Finding √242 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 | 242 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 16.0000000000 | 15.1250000000 | 15.5625000000 | 2 |
| 2 | 15.5625000000 | 15.5502008032 | 15.5563504016 | 5 |
| 3 | 15.5563504016 | 15.5563479706 | 15.5563491861 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √242 = 15.5563491861 to every decimal shown.
√242 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √242 the pattern is [15; 1, 1, 3, 1, 14, 1, 3, 1, 1, 30] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √242 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 | 5.6 × 10⁻¹ |
| 16/1 | 16.0000000000 | 4.4 × 10⁻¹ |
| 31/2 | 15.5000000000 | 5.6 × 10⁻² |
| 109/7 | 15.5714285714 | 1.5 × 10⁻² |
| 140/9 | 15.5555555556 | 7.9 × 10⁻⁴ |
| 2,069/133 | 15.5563909774 | 4.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 242y² = 1. Its smallest solution in positive whole numbers is x = 19,601, y = 1,260.
√242 in geometry and everyday measurements
- A square patio or deck of 242 square feet is about 15.56 ft (15 ft 7 in) on each side, so edging all the way around takes 4 × √242 ≈ 62.2 ft.
- 242 = 11² + 11², so by the Pythagorean theorem √242 is the diagonal of a 11 × 11 rectangle — and the distance between the points (0, 0) and (11, 11) on a grid.
- Since √242 = 11√2, a length of √242 is exactly 11 copies of the length √2 laid end to end.
Square roots near √242 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √239 | √239 | 15.4596 | No |
| √240 | 4√15 | 15.4919 | No |
| √241 | √241 | 15.5242 | No |
| √242 | 11√2 | 15.5563 | No |
| √243 | 9√3 | 15.5885 | No |
| √244 | 2√61 | 15.6205 | No |
| √245 | 7√5 | 15.6525 | No |
- The cube root of 242 is about 6.231680.
- Four times the radicand doubles the root: √968 = 2 × √242 ≈ 31.112698.
Frequently asked questions
What is the square root of 242?
The square root of 242 is 11√2 in simplest radical form, which is about 15.5563491861. The negative root, −15.556349, also squares to 242.
Is the square root of 242 rational or irrational?
Irrational. 242 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 √242 be simplified?
Yes. The largest perfect square dividing 242 is 121, so √242 = √121 × √2 = 11√2.
What is √242 rounded to two decimal places?
√242 ≈ 15.56 to two decimal places (15.6 to one, 15.556 to three). Check: 15.56² = 242.1136, close to 242.