Square Root of 216
2026-02-28 10:34 Diff
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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in fields such as vehicle design, finance, etc. Here, we will discuss the square root of 216.

What is the Square Root of 216?

The square root is the inverse of the square of the number. 216 is not a perfect square. The square root of 216 is expressed in both radical and exponential form. In the radical form, it is expressed as √216, whereas (216)^(1/2) in the exponential form. √216 ≈ 14.69694, 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 216

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 216 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 216 is broken down into its prime factors.

Step 1: Finding the prime factors of 216 Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 3: 2^3 x 3^3

Step 2: Now we found the prime factors of 216. The second step is to make pairs of those prime factors. Since 216 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 216 using prime factorization is impossible.

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Square Root of 216 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 using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 216, we need to group it as 16 and 2.

Step 2: Now we need to find n whose square is 2. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 2. Now the quotient is 1; after subtracting 1, the remainder is 1.

Step 3: Now let us bring down 16, which is the new dividend. Add the old divisor with the same number 1 + 1; we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 116; let us consider n as 4; now 2 x 4 x 4 = 96.

Step 6: Subtract 116 from 96; the difference is 20, and the quotient is 14.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2000.

Step 8: Now we need to find the new divisor that is 146, because 1464 x 4 = 1964.

Step 9: Subtracting 1964 from 2000, we get the result 36.

Step 10: Now the quotient is 14.6.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero.

So the square root of √216 is approximately 14.69.

Square Root of 216 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 216 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √216. The smallest perfect square less than 216 is 196, and the largest perfect square greater than 216 is 225. √216 falls somewhere between 14 and 15.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula, (216 - 196) ÷ (225 - 196) = 20 ÷ 29 ≈ 0.6897. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 14 + 0.69 ≈ 14.69, so the square root of 216 is approximately 14.69.

Common Mistakes and How to Avoid Them in the Square Root of 216

Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping long division steps, etc. Now let us look at a few of those mistakes that students tend to make in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √216?

Okay, lets begin

The area of the square is approximately 216 square units.

Explanation

The area of the square = side^2.

The side length is given as √216.

Area of the square = (√216) x (√216) = 14.69 × 14.69 ≈ 216.

Therefore, the area of the square box is approximately 216 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 216 square feet is built; if each of the sides is √216, what will be the square feet of half of the building?

Okay, lets begin

108 square feet

Explanation

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

Dividing 216 by 2 = we get 108.

So half of the building measures 108 square feet.

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Problem 3

Calculate √216 × 4.

Okay, lets begin

Approximately 58.79

Explanation

The first step is to find the square root of 216, which is approximately 14.69.

The second step is to multiply 14.69 by 4. So 14.69 × 4 ≈ 58.79.

Well explained 👍

Problem 4

What will be the square root of (200 + 16)?

Okay, lets begin

The square root is approximately 14.69.

Explanation

To find the square root, we need to find the sum of (200 + 16).

200 + 16 = 216, and then √216 ≈ 14.69.

Therefore, the square root of (200 + 16) is approximately ±14.69.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √216 units and the width ‘w’ is 40 units.

Okay, lets begin

The perimeter of the rectangle is approximately 109.38 units.

Explanation

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

Perimeter = 2 × (√216 + 40) = 2 × (14.69 + 40) = 2 × 54.69 ≈ 109.38 units.

Well explained 👍

FAQ on Square Root of 216

1.What is √216 in its simplest form?

The prime factorization of 216 is 2 x 2 x 2 x 3 x 3 x 3, so the simplest form of √216 is √(2^3 x 3^3).

2.Mention the factors of 216.

Factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.

3.Calculate the square of 216.

We get the square of 216 by multiplying the number by itself, that is 216 x 216 = 46656.

4.Is 216 a prime number?

5.216 is divisible by?

216 has many factors; those are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.

Important Glossaries for the Square Root of 216

  • Square root: A square root is the inverse of a square. For example: 4^2 = 16 and the inverse of the square is the square root, that is √16 = 4.
  • 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.
  • Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
  • Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.
  • Long division method: A method used to find the square root of non-perfect squares through a series of division steps.

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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.

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