Square Root of 3
Calculate √3 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √3
The square root of 3 is approximately 1.7320508076. Since 3 is not a perfect square, its square root is an irrational number. We know that √1 = 1 and √4 = 2, so √3 falls between 1 and 2. To estimate manually: start with 1.7, then refine using the Babylonian method — guess a value, divide 3 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √1 = 1 and √4 = 2, a good starting guess is 1.5.
- Divide 3 by your guess: 3 ÷ 1.5 = 2.0000
- Average the result with your guess: (1.5 + 2.0000) ÷ 2 = 1.750000
- Repeat with the new estimate until you reach the desired precision.
√3 ≈ 1.7320508076 Properties of √3
Perfect Square?
No
3 is not a perfect square. Its root is irrational.
Decimal Value
1.7320508076
Rounded to 10 decimal places.
Square Root Symbol
√3
Also written as 3^(1/2) or 3^0.5 in programming.
Frequently Asked Questions
What is the square root of 3?
The square root of 3 is approximately 1.7320508076. This is an irrational number.
Is 3 a perfect square?
No, 3 is not a perfect square. Its square root (1.7320508076...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √3 by hand?
To calculate √3 by hand, use the Babylonian method: (1) Make an initial guess — since √1 = 1 and √4 = 2, start with 1.5. (2) Divide 3 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √3 in simplified radical form?
√3 is already in its simplest radical form. The number 3 has no perfect square factors other than 1.