The square root of 3 is one of the most important numbers in mathematics. Also known as Theodorus’s constant, it appears in geometry, trigonometry, and engineering.
The exact value of √3 = 1.73205080…
Want instant results? Use our free square root calculator to calculate any number in seconds!
What is the Square Root of 3?
Simply put, the square root of 3 is the number that, when multiplied by itself, gives 3.
- 1.732 × 1.732 ≈ 3 ✅
| Form | Value |
|---|---|
| Decimal Form | 1.73205080… |
| Simplified Radical | √3 |
| Polar Form | 1.7321 ∠ 0° |
| Perfect Square? | ❌ No |
Is √3 Rational or Irrational?
The square root of 3 is irrational. This means:
- Its decimal never ends
- It cannot be written as a fraction
- Full value: 1.7320508075688772…
According to Wikipedia, √3 is also known as Theodorus’s constant, named after the Greek mathematician Theodorus of Cyrene.
How to Calculate Square Root of 3?
Method 1 — Use a Calculator (Easiest)
The fastest way is our free square root calculator. Just enter 3 and get:
- Decimal form
- Simplified radical
- Step-by-step working
Method 2 — Estimation Method
This is the easiest manual method:
- Find two perfect squares √3 is between
- 1² = 1 and 2² = 4 → so √3 is between 1 and 2
- Try 1.7² = 2.89 (close!)
- Try 1.73² = 2.9929 (very close!)
- Try 1.732² = 2.999824 ≈ 3 ✅
Method 3 — Long Division Method
For more precision:
- Write 3 as 3.000000
- Largest square ≤ 3 → 1² = 1
- Remainder = 2, bring down 00 → 200
- Double quotient = 2, find X: 27 × 7 = 189
- Continue → √3 ≈ 1.732050…
You can verify your answer using our square roots list for reference.
Step by Step — √3 Calculation
- Input: 3
- Apply √: √3
- Decimal: 1.732051
- Simplified: √3 (already simplified)
- Polar form: 1.7321 ∠ 0°
- Perfect square: No
Real Life Uses of √3
You might be surprised — √3 appears everywhere!
- Geometry — Height of equilateral triangle = (√3/2) × side
- Trigonometry — sin(60°) = √3/2 = 0.866025
- Engineering — 3-phase electrical power uses √3
- Architecture — Building proportions and design
- Physics — Many physics formulas include √3
According to Math is Fun, irrational numbers like √3 are extremely important in mathematics and science.
√3 in Trigonometry
√3 is very common in trigonometry:
| Function | Value |
|---|---|
| sin(60°) | √3/2 = 0.866025 |
| cos(30°) | √3/2 = 0.866025 |
| tan(60°) | √3 = 1.732051 |
| cot(30°) | √3 = 1.732051 |
Square Root of 3 vs Other Numbers
| Number | Square Root | Rational? |
|---|---|---|
| 1 | 1.000000 | ✅ Yes |
| 2 | 1.414214 | ❌ No |
| 3 | 1.732051 | ❌ No |
| 4 | 2.000000 | ✅ Yes |
| 9 | 3.000000 | ✅ Yes |
Want to see more values? Check our complete square roots list from 1 to 200.
Tips to Remember √3
- √3 ≈ 1.732 — easy to remember!
- Think of it as “1.7 something”
- It is slightly more than 1.7
- Always between 1 and 2
Frequently Asked Questions
What is the square root of 3?
The square root of 3 is 1.73205080… It is an irrational number whose decimal never ends or repeats.
Is 3 a perfect square?
No, 3 is not a perfect square. Perfect squares are 1, 4, 9, 16, 25… Check our perfect squares page for full list.
What is √3 to 10 decimal places?
√3 = 1.7320508075
Can √3 be simplified?
No, √3 cannot be simplified. Since 3 is a prime number, it has no perfect square factors.
What is √3 × √3?
√3 × √3 = 3 — when you multiply a square root by itself, you always get the original number!
Is √3 bigger than √2?
Yes! √3 = 1.732051 is bigger than √2 = 1.414214.
Conclusion
The square root of 3 is approximately 1.732051. It is an irrational number that appears in geometry, trigonometry, and engineering.
Key takeaways:
- √3 = 1.73205080…
- It is irrational — decimal never ends
- Cannot be simplified further
- Used in sin(60°), cos(30°), tan(60°)
For instant results with full step-by-step working, try our free square root calculator — no signup required!
