√237 at a glance
- Exact value
- √237
- Decimal (10 places)
- 15.3948043183
- Rounded
- 15.4 · 15.39 · 15.395
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.394804
- Prime factorization
- 3 × 79
- Cube root
- 6.188463
How to simplify √237
The prime factorization of 237 is 3 × 79. Every prime appears only once, so there is no pair to bring outside the radical — √237 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 237, 3 and 79 appear an odd number of times, so √237 is irrational and 15.3948043183 is a rounded value.
Where √237 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √237 lies between 15 and 16. 237 is 12 above 225 and 19 below 256, so the root is closer to 15.
- Straight line between 225 and 256: 15.3871 (0.05% low)
- Tangent from 15, i.e. 15 + 12 ÷ 30: 15.4000 (0.03% high)
- Tangent from 16, i.e. 16 − 19 ÷ 32: 15.4063 (0.07% high)
For √237 the tangent at 15 wins, missing by only 0.0052. Tangent estimates shine when the number sits close to a perfect square — here 237 is just 12 above 225.
Finding √237 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 237: following the tangent line down to zero simplifies to averaging x with 237 ÷ x.
Start from the nearest whole number, 15 (15² = 225):
| Step | Guess x | 237 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 15.0000000000 | 15.8000000000 | 15.4000000000 | 2 |
| 2 | 15.4000000000 | 15.3896103896 | 15.3948051948 | 6 |
| 3 | 15.3948051948 | 15.3948034419 | 15.3948043183 | 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 √237 = 15.3948043183 to every decimal shown.
√237 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √237 the pattern is [15; 2, 1, 1, 7, 10, 7, 1, 1, 2, 30] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √237 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 | 3.9 × 10⁻¹ |
| 31/2 | 15.5000000000 | 1.1 × 10⁻¹ |
| 46/3 | 15.3333333333 | 6.1 × 10⁻² |
| 77/5 | 15.4000000000 | 5.2 × 10⁻³ |
| 585/38 | 15.3947368421 | 6.7 × 10⁻⁵ |
| 5,927/385 | 15.3948051948 | 8.8 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 237y² = 1. Its smallest solution in positive whole numbers is x = 228,151, y = 14,820.
√237 in geometry and everyday measurements
- A square patio or deck of 237 square feet is about 15.39 ft (15 ft 5 in) on each side, so edging all the way around takes 4 × √237 ≈ 61.6 ft.
- 237 is not a sum of two whole-number squares — the prime factor 3 and 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √237 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 13 box, because 2² + 8² + 13² = 237.
Square roots near √237 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √234 | 3√26 | 15.2971 | No |
| √235 | √235 | 15.3297 | No |
| √236 | 2√59 | 15.3623 | No |
| √237 | √237 | 15.3948 | No |
| √238 | √238 | 15.4272 | No |
| √239 | √239 | 15.4596 | No |
| √240 | 4√15 | 15.4919 | No |
- The cube root of 237 is about 6.188463.
- Four times the radicand doubles the root: √948 = 2 × √237 ≈ 30.789609.
Frequently asked questions
What is the square root of 237?
The square root of 237 is √237, about 15.3948043183. The negative root, −15.394804, also squares to 237.
Is the square root of 237 rational or irrational?
Irrational. 237 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 √237 be simplified?
No. 237 = 3 × 79 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √237 rounded to two decimal places?
√237 ≈ 15.39 to two decimal places (15.4 to one, 15.395 to three). Check: 15.39² = 236.8521, close to 237.