√802 at a glance
- Exact value
- √802
- Decimal (10 places)
- 28.3196045170
- Rounded
- 28.3 · 28.32 · 28.320
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.319605
- Prime factorization
- 2 × 401
- Cube root
- 9.290907
How to simplify √802
The prime factorization of 802 is 2 × 401. Every prime appears only once, so there is no pair to bring outside the radical — √802 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 802, 2 and 401 appear an odd number of times, so √802 is irrational and 28.3196045170 is a rounded value.
Where √802 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √802 lies between 28 and 29. 802 is 18 above 784 and 39 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.3158 (0.01% low)
- Tangent from 28, i.e. 28 + 18 ÷ 56: 28.3214 (0.01% high)
- Tangent from 29, i.e. 29 − 39 ÷ 58: 28.3276 (0.03% high)
For √802 the tangent at 28 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 802 is just 18 above 784.
Finding √802 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, 28 (28² = 784):
| Step | Guess x | 802 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.6428571429 | 28.3214285714 | 2 |
| 2 | 28.3214285714 | 28.3177805801 | 28.3196045758 | 7 |
| 3 | 28.3196045758 | 28.3196044583 | 28.3196045170 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √802 = 28.3196045170 to every decimal shown.
√802 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √802 the pattern is [28; 3, 7, 1, 3, 6, 28, 6, 3, 1, 7, 3, 56] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √802 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 3.2 × 10⁻¹ |
| 85/3 | 28.3333333333 | 1.4 × 10⁻² |
| 623/22 | 28.3181818182 | 1.4 × 10⁻³ |
| 708/25 | 28.3200000000 | 4.0 × 10⁻⁴ |
| 2,747/97 | 28.3195876289 | 1.7 × 10⁻⁵ |
| 17,190/607 | 28.3196046129 | 9.6 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 802y² = 1. Its smallest solution in positive whole numbers is x = 295,496,099, y = 10,434,330.
√802 in geometry and everyday measurements
- 802 square feet is 74.5 m². Laid out as a square — a small house footprint or a lot — it is about 28.32 ft (28 ft 4 in) on a side.
- 802 = 19² + 21², so by the Pythagorean theorem √802 is the diagonal of a 19 × 21 rectangle — and the distance between the points (0, 0) and (19, 21) on a grid.
Square roots near √802 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √799 | √799 | 28.2666 | No |
| √800 | 20√2 | 28.2843 | No |
| √801 | 3√89 | 28.3019 | No |
| √802 | √802 | 28.3196 | No |
| √803 | √803 | 28.3373 | No |
| √804 | 2√201 | 28.3549 | No |
| √805 | √805 | 28.3725 | No |
- The cube root of 802 is about 9.290907.
- Squaring undoes the root: (√802)² = 802, while 802² = 643,204 — the number whose square root is 802.
Frequently asked questions
What is the square root of 802?
The square root of 802 is √802, about 28.3196045170. The negative root, −28.319605, also squares to 802.
Is the square root of 802 rational or irrational?
Irrational. 802 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √802 be simplified?
No. 802 = 2 × 401 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √802 rounded to two decimal places?
√802 ≈ 28.32 to two decimal places (28.3 to one, 28.320 to three). Check: 28.32² = 802.0224, close to 802.