√894 at a glance
- Exact value
- √894
- Decimal (10 places)
- 29.8998327755
- Rounded
- 29.9 · 29.90 · 29.900
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.899833
- Prime factorization
- 2 × 3 × 149
- Cube root
- 9.633391
How to simplify √894
The prime factorization of 894 is 2 × 3 × 149. Every prime appears only once, so there is no pair to bring outside the radical — √894 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 894, 2, 3 and 149 appear an odd number of times, so √894 is irrational and 29.8998327755 is a rounded value.
Where √894 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √894 lies between 29 and 30. 894 is 53 above 841 and 6 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.8983 (0.01% low)
- Tangent from 29, i.e. 29 + 53 ÷ 58: 29.9138 (0.05% high)
- Tangent from 30, i.e. 30 − 6 ÷ 60: 29.9000 (0% high)
For √894 the tangent at 30 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 894 is just 6 below 900.
Finding √894 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, 30 (30² = 900):
| Step | Guess x | 894 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.8000000000 | 29.9000000000 | 3 |
| 2 | 29.9000000000 | 29.8996655518 | 29.8998327759 | 9 |
| 3 | 29.8998327759 | 29.8998327750 | 29.8998327755 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √894 = 29.8998327755 to every decimal shown.
√894 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √894 the pattern is [29; 1, 8, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √894 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 9.0 × 10⁻¹ |
| 30/1 | 30.0000000000 | 1.0 × 10⁻¹ |
| 269/9 | 29.8888888889 | 1.1 × 10⁻² |
| 299/10 | 29.9000000000 | 1.7 × 10⁻⁴ |
| 17,611/589 | 29.8998302207 | 2.6 × 10⁻⁶ |
| 17,910/599 | 29.8998330551 | 2.8 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 894y² = 1. Its smallest solution in positive whole numbers is x = 299, y = 10.
√894 in geometry and everyday measurements
- 894 square feet is 83.1 m². Laid out as a square — a small house footprint or a lot — it is about 29.9 ft (29 ft 11 in) on a side.
- 894 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √894 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 29 box, because 2² + 7² + 29² = 894.
Square roots near √894 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √891 | 9√11 | 29.8496 | No |
| √892 | 2√223 | 29.8664 | No |
| √893 | √893 | 29.8831 | No |
| √894 | √894 | 29.8998 | No |
| √895 | √895 | 29.9166 | No |
| √896 | 8√14 | 29.9333 | No |
| √897 | √897 | 29.9500 | No |
- The cube root of 894 is about 9.633391.
- Squaring undoes the root: (√894)² = 894, while 894² = 799,236 — the number whose square root is 894.
Frequently asked questions
What is the square root of 894?
The square root of 894 is √894, about 29.8998327755. The negative root, −29.899833, also squares to 894.
Is the square root of 894 rational or irrational?
Irrational. 894 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √894 be simplified?
No. 894 = 2 × 3 × 149 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √894 rounded to two decimal places?
√894 ≈ 29.90 to two decimal places (29.9 to one, 29.900 to three). Check: 29.90² = 894.01, close to 894.