Square Root of 1/3
2026-02-28 06:08 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-1/3

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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of 1/3.

What is the Square Root of 1/3?

The square root is the inverse of the square of the number. 1/3 is not a perfect square. The square root of 1/3 is expressed in both radical and exponential form. In the radical form, it is expressed as √(1/3), whereas (1/3)^(1/2) in the exponential form. The square root of 1/3 is approximately 0.57735, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 1/3

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division method and approximation method are used. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 1/3 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root of 1/3 using the long division method, step by step:

Step 1: To begin with, we consider 1/3, which is 0.333...

Step 2: Estimate a number close to 0.333... whose square is less than or equal to this number. We start with 0.5 since 0.5^2 = 0.25.

Step 3: Improve the approximation by using the long division method to find a more accurate value.

Step 4: Continue the long division steps to find more precise decimal places.

The square root of 1/3 is approximately 0.57735.

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Square Root of 1/3 by Approximation Method

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1/3 using the approximation method.

Step 1: Now we have to find the closest perfect squares around 1/3. The closest perfect squares to 1/3 are 0.25 (which is 0.5^2) and 0.5625 (which is 0.75^2). The square root of 1/3 falls somewhere between 0.5 and 0.75.

Step 2: Apply the formula for linear approximation between these two points. Using the approximation method, we conclude the square root of 1/3 is approximately 0.57735.

Common Mistakes and How to Avoid Them in the Square Root of 1/3

Students make mistakes while finding square roots, such as forgetting about the negative square root or skipping steps in methods. Now let us look at a few of those mistakes that students tend to make in detail.

Problem 1

Can you help Max find the area of a square box if its side length is given as √(1/6)?

Okay, lets begin

The area of the square is approximately 0.1667 square units.

Explanation

The area of the square = side^2.

The side length is given as √(1/6).

Area of the square = (√(1/6))^2 = 1/6 ≈ 0.1667.

Therefore, the area of the square box is approximately 0.1667 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 1/3 square feet is built; if each of the sides is √(1/3), what will be the square feet of half of the building?

Okay, lets begin

0.1667 square feet

Explanation

We can just divide the given area by 2 as the building is square-shaped.

Dividing 1/3 by 2 = 1/6 ≈ 0.1667.

So half of the building measures approximately 0.1667 square feet.

Well explained 👍

Problem 3

Calculate √(1/3) × 5.

Okay, lets begin

Approximately 2.88675

Explanation

The first step is to find the square root of 1/3, which is approximately 0.57735.

The second step is to multiply 0.57735 by 5.

So, 0.57735 × 5 ≈ 2.88675.

Well explained 👍

Problem 4

What will be the square root of (1/6 + 1/12)?

Okay, lets begin

Approximately 0.6455

Explanation

To find the square root, first find (1/6 + 1/12).

1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4.

The square root of 1/4 is 1/2 = 0.5.

Therefore, the square root of (1/6 + 1/12) is 0.5.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √(1/3) units and the width ‘w’ is 1 unit.

Okay, lets begin

Approximately 3.1547 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√(1/3) + 1) = 2 × (0.57735 + 1) ≈ 2 × 1.57735 ≈ 3.1547 units.

Well explained 👍

FAQ on Square Root of 1/3

1.What is √(1/3) in its simplest form?

The simplest form of √(1/3) is √(1)/√(3), which can be rationalized to 1/√(3) × √(3)/√(3) = √(3)/3.

2.Is 1/3 an irrational number?

3.Calculate the square of 1/3.

The square of 1/3 is (1/3) × (1/3) = 1/9.

4.Is 1/3 a prime number?

5.What is the reciprocal of 1/3?

The reciprocal of 1/3 is 3.

Important Glossaries for the Square Root of 1/3

  • Square root: A square root is the inverse of a square. Example: 2^2 = 4 and the inverse of the square is the square root, √4 = 2.
  • Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
  • Rational number: A rational number can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero.
  • Reciprocal: The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 1/3 is 3.
  • Approximation: Approximation is the process of finding a value that is close to but not exactly equal to the actual value, often used for irrational numbers.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.