√412 at a glance
- Exact value
- 2√103
- Decimal (10 places)
- 20.2977831302
- Rounded
- 20.3 · 20.30 · 20.298
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.297783
- Prime factorization
- 2² × 103
- Cube root
- 7.441019
How to simplify √412
Look for the largest perfect square that divides 412. Here it is 4 (2²), because 412 = 4 × 103 and 103 has no square factor left:
The prime factorization tells the same story: 412 = 2² × 103. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 103 stays inside.
Check: (2√103)² = 2² × 103 = 4 × 103 = 412. As a decimal, 2√103 = 2 × 10.1488915651 ≈ 20.2977831302.
Where √412 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √412 lies between 20 and 21. 412 is 12 above 400 and 29 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.2927 (0.03% low)
- Tangent from 20, i.e. 20 + 12 ÷ 40: 20.3000 (0.01% high)
- Tangent from 21, i.e. 21 − 29 ÷ 42: 20.3095 (0.06% high)
For √412 the tangent at 20 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 412 is just 12 above 400.
Finding √412 with the Babylonian method
Picture a rectangle with an area of 412 and one side x; the other side must be 412 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √412.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 412 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.6000000000 | 20.3000000000 | 2 |
| 2 | 20.3000000000 | 20.2955665025 | 20.2977832512 | 6 |
| 3 | 20.2977832512 | 20.2977830091 | 20.2977831302 | 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 √412 = 20.2977831302 to every decimal shown.
√412 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √412 the pattern is [20; 3, 2, 1, 3, 1, 4, 3, 2, 13, 10, 13, 2, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √412 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 |
|---|---|---|
| 20/1 | 20.0000000000 | 3.0 × 10⁻¹ |
| 61/3 | 20.3333333333 | 3.6 × 10⁻² |
| 142/7 | 20.2857142857 | 1.2 × 10⁻² |
| 203/10 | 20.3000000000 | 2.2 × 10⁻³ |
| 751/37 | 20.2972972973 | 4.9 × 10⁻⁴ |
| 954/47 | 20.2978723404 | 8.9 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 412y² = 1. Its smallest solution in positive whole numbers is x = 103,537,981,567, y = 5,100,950,232.
√412 in geometry and everyday measurements
- A square garage floor of 412 square feet measures about 20.3 ft (20 ft 4 in) per side, and its corner-to-corner diagonal is √824 ≈ 28.7 ft.
- 412 is not a sum of two whole-number squares — the prime factor 103 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √412 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √412 as its space diagonal.
- Since √412 = 2√103, a length of √412 is exactly 2 copies of the length √103 laid end to end.
Square roots near √412 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √409 | √409 | 20.2237 | No |
| √410 | √410 | 20.2485 | No |
| √411 | √411 | 20.2731 | No |
| √412 | 2√103 | 20.2978 | No |
| √413 | √413 | 20.3224 | No |
| √414 | 3√46 | 20.3470 | No |
| √415 | √415 | 20.3715 | No |
- The cube root of 412 is about 7.441019.
- Because 412 = 4 × 103, the root is twice √103: 2 × 10.148892 ≈ 20.297783.
Frequently asked questions
What is the square root of 412?
The square root of 412 is 2√103 in simplest radical form, which is about 20.2977831302. The negative root, −20.297783, also squares to 412.
Is the square root of 412 rational or irrational?
Irrational. 412 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √412 be simplified?
Yes. The largest perfect square dividing 412 is 4, so √412 = √4 × √103 = 2√103.
What is √412 rounded to two decimal places?
√412 ≈ 20.30 to two decimal places (20.3 to one, 20.298 to three). Check: 20.30² = 412.09, close to 412.