Value of Root 3

Understanding the value of the square root of 3 is a basic concept in mathematics. This is especially true when working with irrational numbers, geometry, and trigonometry. The square root of 3, or √3, is common in mathematical problems, particularly those involving equilateral triangles, 30°-60°-90° triangles, and various engineering tasks.  

This guide explores the value of the root 3, providing explanations, methods to calculate it manually, and a clear breakdown using a square root table and the square root using the long division method

 

Table of Contents

• Root 3 Value in Maths
• How to Find the Value of Root 3?
• Important Key Points To Remember
• Table of Square Root
• Practical Uses of √3 Value
• Conclusion
• FAQs On Value of Root 3

 

Root 3 Value in Maths  

In maths, we often need to calculate the square root of numbers. Some numbers are perfect squares (like 4, 9, 16), and their roots can be found easily by factorisation. But some numbers are non-perfect squares (like 2, 3, 5). Their square roots are not whole numbers, so we use the long division method to find them.

Since 3 is a non-perfect square number, we will calculate the value of √3 using the long division method.

In mathematics, the value of the root 3 is the positive number that, when multiplied by itself, equals 3. Mathematically:  

3×3=3

The square root of 3 is an, which means it cannot be expressed as a simple fraction. Its decimal representation goes on infinitely without repeating. The approximate value of √3 is:  

3≈1.7320508075

This value is often used in trigonometry. For instance:  

• In a 30°-60°-90° triangle, the ratio of the sides includes the square root of 3.  

• The tangent of 60° equals the value of root 3.  

• The height of an equilateral triangle with side length ‘a’ is (√3/2) × a. 

 

How to Find the Value of Root 3?  

You might wonder how to find the square root of 3 manually or without a calculator. Since 3 is not a perfect square number, we use the long division method to find its square root.

Steps:

1. Write 3 as 3.000000… and group the digits in pairs: (3)(00)(00)(00)…

2. Find the largest number whose square is ≤ 3.

   • 1 × 1 = 1, but 2 × 2 = 4 (too big).

   • So the first digit is 1.
     subtract 3 - 1² = 2, bring down 00 → remainder 200.

3. Double the divisor (1 → 2).
   find a digit X such that (20 + X) × X ≤ 200.

   • try X = 7 → 27 × 7 = 189. fits.
     quotient becomes 1.7.
     subtract 200 - 189 = 11, bring down 00 → remainder 1100.

4. Double the quotient so far (17 → 34).
   find X such that (340 + X) × X ≤ 1100.

   • try X = 3 → 343 × 3 = 1029. fits.
     quotient becomes 1.73.
     subtract 1100 - 1029 = 71, bring down 00 → remainder 7100.

5. Double the quotient so far (173 → 346).
   find X such that (3460 + X) × X ≤ 7100.

   • try X = 2 → 3462 × 2 = 6924. fits.
     quotient becomes 1.732.

By continuing this process, we get:

$\sqrt{3 =1.73205080757…}$

Final result:
The value of the root 3 is approximately:

$\sqrt{3 ​≈1.732}$

(In most math problems, we round √3 to three decimal places as 1.732).

 

Important Key Points To Remember

Here are some key points about the value of √3:

• √3 means “a number which, when multiplied by itself, equals 3.”

• It is not a perfect square, so its value cannot be a whole number or a simple fraction.

• The exact value of √3 is an irrational number (its decimals go on forever without repeating).

• using the long division method, we find:

  $\sqrt{3=1.73205080757.....}$

• For most calculations, we use the rounded value √3 ≈ 1.732 (correct up to three decimal places).

• √3 is very important in mathematics, especially in geometry and trigonometry (example: the height of an equilateral triangle involves √3).

 

Table of Square Roots  

The square root table below gives approximate values of common square roots. It is especially helpful for solving problems manually.  

 

As shown in the square root table, the value of root 3 falls between √2 and √4. This table offers a quick reference for commonly used √ values, with √3 value among the most frequently encountered.

 

Practical Uses of √3 Value  

The value of root 3 appears in many real-life applications:  

• Calculating heights and lengths in architecture  

• Electrical engineering (three-phase current systems)  

• Trigonometry (sine and tangent for 60°)  

• Physics (vector analysis) 

Knowing how to find the square root of 3 manually using estimation or the long division method is beneficial for both academic and practical problem-solving.  

 

Conclusion  

The value of root 3 is an important mathematical constant, approximately equal to 1.732. Since √3 is an irrational number, its value cannot be expressed as a simple fraction and its decimal expansion goes on infinitely without repeating. √3 appears often in geometry (equilateral triangles, 30°-60°-90° triangles), trigonometry (tan 60° = √3), and in practical fields like engineering, architecture, and physics. Therefore, learning how to find and use the value of √3 is a useful skill for both academic studies and real-life problem-solving.

 

Frequently Asked Questions on Square Roots  

1. What is √3 of value?  

Answer: The value of √3 (square root of 3) is approximately 1.732. It is an irrational number and cannot be expressed exactly as a simple fraction.  

 

2. What is √2 of value?

Answer: The value of √2 (square root of 2) is approximately 1.414. Like √3, it is also an irrational number.  

 

3. How to solve √3? 

Answer: To solve or estimate √3 manually, you can use methods like long division, approximation, or a calculator. One way to estimate is by trying successive squares:  

   For example,  

   1.7² = 2.89  

   1.73² = 2.9929  

   1.732² ≈ 3.0001 → So √3 ≈ 1.732  

 

4. How to find √5 value? 

Answer: The value of √5 (square root of 5) is approximately 2.236. You can find it using a calculator or by using estimation methods similar to those used for √3.  

 

Learn how to calculate and apply the value of root 3 with step-by-step guidance from Orchids The International School!