√462 at a glance
- Exact value
- √462
- Decimal (10 places)
- 21.4941852602
- Rounded
- 21.5 · 21.49 · 21.494
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.494185
- Prime factorization
- 2 × 3 × 7 × 11
- Cube root
- 7.730614
How to simplify √462
The prime factorization of 462 is 2 × 3 × 7 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √462 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 462, 2, 3, 7 and 11 appear an odd number of times, so √462 is irrational and 21.4941852602 is a rounded value.
Where √462 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √462 lies between 21 and 22. 462 is 21 above 441 and 22 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.4884 (0.03% low)
- Tangent from 21, i.e. 21 + 21 ÷ 42: 21.5000 (0.03% high)
- Tangent from 22, i.e. 22 − 22 ÷ 44: 21.5000 (0.03% high)
For √462 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √462 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, 21 (21² = 441):
| Step | Guess x | 462 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 22.0000000000 | 21.5000000000 | 2 |
| 2 | 21.5000000000 | 21.4883720930 | 21.4941860465 | 6 |
| 3 | 21.4941860465 | 21.4941844739 | 21.4941852602 | 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 √462 = 21.4941852602 to every decimal shown.
√462 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √462 the pattern is [21; 2, 42] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √462 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 4.9 × 10⁻¹ |
| 43/2 | 21.5000000000 | 5.8 × 10⁻³ |
| 1,827/85 | 21.4941176471 | 6.8 × 10⁻⁵ |
| 3,697/172 | 21.4941860465 | 7.9 × 10⁻⁷ |
| 157,101/7,309 | 21.4941852511 | 9.1 × 10⁻⁹ |
| 317,899/14,790 | 21.4941852603 | 1.1 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 462y² = 1. Its smallest solution in positive whole numbers is x = 43, y = 2.
√462 in geometry and everyday measurements
- A square garage floor of 462 square feet measures about 21.49 ft (21 ft 6 in) per side, and its corner-to-corner diagonal is √924 ≈ 30.4 ft.
- 462 is not a sum of two whole-number squares — the prime factor 3, 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √462 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 19 box, because 1² + 10² + 19² = 462.
Square roots near √462 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √459 | 3√51 | 21.4243 | No |
| √460 | 2√115 | 21.4476 | No |
| √461 | √461 | 21.4709 | No |
| √462 | √462 | 21.4942 | No |
| √463 | √463 | 21.5174 | No |
| √464 | 4√29 | 21.5407 | No |
| √465 | √465 | 21.5639 | No |
- The cube root of 462 is about 7.730614.
- Squaring undoes the root: (√462)² = 462, while 462² = 213,444 — the number whose square root is 462.
Frequently asked questions
What is the square root of 462?
The square root of 462 is √462, about 21.4941852602. The negative root, −21.494185, also squares to 462.
Is the square root of 462 rational or irrational?
Irrational. 462 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √462 be simplified?
No. 462 = 2 × 3 × 7 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √462 rounded to two decimal places?
√462 ≈ 21.49 to two decimal places (21.5 to one, 21.494 to three). Check: 21.49² = 461.8201, close to 462.