√163 at a glance
- Exact value
- √163
- Decimal (10 places)
- 12.7671453348
- Rounded
- 12.8 · 12.77 · 12.767
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.767145
- Prime factorization
- 163
- Cube root
- 5.462556
How to simplify √163
163 is a prime number, so its only factors are 1 and 163. There is no perfect-square factor to pull out, which means √163 is already in its simplest radical form.
The square root of any prime is irrational. If √163 were a fraction a/b in lowest terms, then a² = 163b², so 163 would divide a — and then 163 would divide b too, contradicting “lowest terms.” That is why the decimal 12.7671453348 is only a rounded value.
Where √163 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √163 lies between 12 and 13. 163 is 19 above 144 and 6 below 169, so the root is closer to 13.
- Straight line between 144 and 169: 12.7600 (0.06% low)
- Tangent from 12, i.e. 12 + 19 ÷ 24: 12.7917 (0.19% high)
- Tangent from 13, i.e. 13 − 6 ÷ 26: 12.7692 (0.02% high)
For √163 the tangent at 13 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 163 is just 6 below 169.
Finding √163 with the Babylonian method
If a guess is too big, 163 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√163) in one step.
Start from the nearest whole number, 13 (13² = 169):
| Step | Guess x | 163 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 12.5384615385 | 12.7692307692 | 2 |
| 2 | 12.7692307692 | 12.7650602410 | 12.7671455051 | 6 |
| 3 | 12.7671455051 | 12.7671451645 | 12.7671453348 | 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 √163 = 12.7671453348 to every decimal shown.
√163 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √163 the pattern is [12; 1, 3, 3, 2, 1, 1, 7, 1, 11, 1, 7, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √163 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 |
|---|---|---|
| 12/1 | 12.0000000000 | 7.7 × 10⁻¹ |
| 13/1 | 13.0000000000 | 2.3 × 10⁻¹ |
| 51/4 | 12.7500000000 | 1.7 × 10⁻² |
| 166/13 | 12.7692307692 | 2.1 × 10⁻³ |
| 383/30 | 12.7666666667 | 4.8 × 10⁻⁴ |
| 549/43 | 12.7674418605 | 3.0 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 163y² = 1. Its smallest solution in positive whole numbers is x = 64,080,026, y = 5,019,135.
√163 in geometry and everyday measurements
- A square room or garden bed covering 163 square feet measures about 12.77 ft (12 ft 9 in) along each wall.
- 163 is not a sum of two whole-number squares — 163 is itself a prime that is one less than a multiple of 4, which rules that out — so √163 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 9 box, because 1² + 9² + 9² = 163.
Square roots near √163 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √160 | 4√10 | 12.6491 | No |
| √161 | √161 | 12.6886 | No |
| √162 | 9√2 | 12.7279 | No |
| √163 | √163 | 12.7671 | No |
| √164 | 2√41 | 12.8062 | No |
| √165 | √165 | 12.8452 | No |
| √166 | √166 | 12.8841 | No |
- The cube root of 163 is about 5.462556.
- Four times the radicand doubles the root: √652 = 2 × √163 ≈ 25.534291.
Frequently asked questions
What is the square root of 163?
The square root of 163 is √163, about 12.7671453348. The negative root, −12.767145, also squares to 163.
Is the square root of 163 rational or irrational?
Irrational. 163 is not a perfect square — it falls between 144 and 169 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √163 be simplified?
No. 163 is prime, so there is no perfect square to take out of the radical.
What is √163 rounded to two decimal places?
√163 ≈ 12.77 to two decimal places (12.8 to one, 12.767 to three). Check: 12.77² = 163.0729, close to 163.