Square Root Tricks
2026-02-28 08:29 Diff
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Last updated on October 29, 2025

Finding square roots can sometimes be a challenging and time-consuming process. But these few techniques can make this process quicker and easier. This article discusses tips that help you find the square root efficiently.

What is Square Root?

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The value that, when multiplied by itself, results in the original number is known as the square root of that number. The square root symbol is √. The square root of a number can be rational (like 6 or 2.5) or irrational (√2 or √5). If the square root of a number is a whole number, then that number is called a perfect square. For example, √25 is 5.
 

Tricks to Find Square Root

The square root of a perfect square number can be found using basic tricks, without relying on long division. One of the helpful methods is to remember the unit digits of the squares for the first ten natural numbers. These tips can help us find the square root of a number when solving complex problems. Given below is a unit digit table that is used to find the square root of the given number.
 

Numbers

Unit Digits of Squares of Numbers 

1 1 2 4 3 9 4 6 5 5 6 6 7 9 8 4 9 1 10 0

How to Find the Square Root of the Large Numbers?

Finding a square root for a larger number like 1,567,865 can be difficult. There are some simple steps, as given below, that can help you solve them easily. 
 

Step 1: Starting from the right, group the digits of the number into pairs of two.
 

Step 2: Look at the last digit of the number. Use the unit digit table of numbers from 1 to 9 to find which numbers can give that last digit when squared. Those are the possible unit digits of the square root.
 

Step 3: Look at the leftmost pair, i.e., the first pair, and find between which two squares this number lies.
 

Step 4: The smaller of those two is the tens' digit of the square root.
 

Step 5: If the unit digit is 5 or 0, then it means the number that we’re looking for also ends with either 5 or 0. If the unit digit is not 0 or 5, use trial and error with the possibilities from Step 3 to find the correct unit digit.
 

Step 6: Try different digits to complete the divisor and multiply. Choose the largest possible digit that, when used, gives a product less than or equal to the number you're dividing in that step. 
 

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Square Root Tricks for 3-Digit Numbers

The square root of a 3-digit number is usually a 2-digit number, but if the number is a perfect square above 961 (like 1024), the square root can be a 3-digit number. The trick to finding the square root of a three-digit number is given below, with a simple example:

Step 1: Group the digits from the right and make pairs of digits. For 729, we group it as 7 | 29 — the rightmost two digits form a pair, and the remaining digit stands alone.

Step 2: Look at the last digit of the number and use the unit table to find the unit digit of the square root of the given number. Here, the last digit is 9, so we have to find the squares that end in 9. 
32 = 9
72 = 49
So, the unit digit of the square root is either 3 or 7.
 

Step 3: Now, look at the leftmost group of digits and find the largest perfect square less than or equal to that number. The square root of that perfect square gives the tens digit of the final answer. Here, the first digit is 7. The perfect squares which are closest to 7 are:
22 = 4
32 = 9
9 is too big so we are using 22 = 4, and √4 = 2.
So, the tens digit is 2.
 

Step 4: Combine the two guesses; the number we’re looking for can be one of those two guesses.
The possible guesses are:
23 (tens digit 2, unit digit 3)
27 (tens digit 2, unit digit 7)
Now check both:
23 × 23 = 529
27 × 27 = 729

So, the square root of 729 is ±27.
 

Square Root Tricks for 4-Digit Numbers

Finding the square root of a 4-digit number follows similar steps as for 3-digit numbers, with a few extra rules due to the larger size. Here's how to do it:

Step 1: Group the digits:
Start from the right and group the digits into pairs of two. 
 

Step 2: Find the nearest perfect square:
Look at the first pair and find the biggest perfect square that is less than or equal to it.
 

Step 3: Find the first digit:
Take the square root of that perfect square, and it is the first digit of the answer.
 

Step 4: Find the remaining digits:
Now look at the second pair of digits and find the next digit of the square root.
 

Step 5: Combine the digits:
Put the digits from step 3 and step 4 together to make the square root guess.
 

Step 6: Check the answer:
Multiply the guesses by itself to see whether it gives the original number. 

Example: Find the square root of 9604


Step 1: Group the digits (96)(04).


Step 2: The first pair is 96. The perfect square less than 96 is 81.


Step 3: The square root of 81 is ±9. So, the first digit is 9.


Step 4: Now check the next pair: 04
The unit digits are:
22 = 4
82 = 64
So the second digit is either 2 or 8.


Step 5: Combine the terms: 
92 × 92 = 8464
98 × 98 = 9604
So the square root of 9604 is ±98.


Step 6: Final check: 98 × 98 = 9604
 

Square Root Tricks for 5 Digit Numbers

Finding the square root of a 5-digit number is difficult, but by using the following trick, you can find it easily.
 

Step 1: Group the numbers:
Start from the right and split the numbers into two parts. The first part should contain 3 digits, and the second part should have 2 digits. For example: 53824 can be grouped as | 5 | 38 | 24.
 

Step 2: Check the last digit:
Look for the last digit, here it is 4. Check the numbers that give that number when squared.
22 = 4
82 = 64
So the last digit of the square root is either 2 or 8.
 

Step 3: Use the first three digits:
Look at the first two groups: 538
Find the closest perfect squares
232 = 529
242 = 576
So, 538 is between 529 and 576. We chose 23 as the first part of the answer, as 242 is big.
 

Step 4: Combine the digits:
Now square the number you found to check if it matches the original number.
232 × 232 = 53824
238 × 238 = 56644
So, the square root of 53824 is ±232.
 

Tips and Tricks to Master Square Root Tricks

Finding the square root of a number, especially a large one, can be tricky if you rely only on standard long-division. Here are some effective tips and tricks for students to master square root tricks. 

  • Know the unit-digit table of squares: Memorize how the unit digit of a square behaves. For example, 1² ends in 1, 2² ends in 4, 3² ends in 9, 4² ends in 6, etc. This helps you immediately narrow down what the unit digit of a root could be.
  • Group digits from the right for large numbers: When dealing with 3, 4 or 5-digit numbers (or more), start from the right and form pairs of two digits (and sometimes a three-digit group for the leftmost part) to ease the square-root process.
  • Use the leftmost group to estimate the tens digit of the root: Look at the leftmost grouped digits and determine between which two perfect squares that group lies. The smaller of those two roots gives the tens (or next) digit of the answer.
  • Combine and refine: Combine the unit-digit hint with the tens-digit estimate, and then refine via trial. Once you have a possible unit digit (from the first table) and an estimated tens/higher digit (from the leftmost group), you form candidate roots and verify which one squares correctly (or nearly correctly).
  • Know the limits while applying the tricks: Use these tricks for speed and estimation, but know their limits. These methods work best when the number is a perfect square (or close to one). For non-perfect squares, these tricks give a good approximation, but you may still need a calculator or formal method for an exact irrational result.

Common Mistakes and How to Avoid Them in Square Root Tricks

Square root tricks are great for mental math and problem-solving, but a small mistake can lead to big errors. Learning about some of the common mistakes beforehand can help us avoid them in the future. 

Real Life Applications of Square Root Tricks

Learning the square root tricks helps us calculate faster, even without a calculator. From measuring things to solving puzzles, square root tricks play an important role in both everyday life and advanced fields like design, science, and engineering. Here are some of the real-life applications of square root tricks.

  • Sports fields: Players and coaches use square roots to figure out the side length of a square field. If a square playing zone has an area of 625 sq. ft, the length of each side is √625 = 25 ft.
  • Maps and navigation: Square root tricks are used in the Pythagorean theorem to find straight line distances between two points, like on maps or GPS. For example, Distance = x2 + y2 (e.g., 32 + 42  = 25 = 5 units).
  • Physics and engineering calculations: Square roots appear in formulas for speed, energy, force, and more. Square root tricks can help in solving them faster. For example, to find the speed from kinetic energy: v = 2Em (e.g., E = 100 J, m = 2 kg  v = 100 = 10 m/s).
  • Video games and computer graphics: Square roots are used in graphics, distance between game characters, or size scaling. Game developers use square root calculations to detect how far a player is from a target.
  • DIY projects and construction estimates: Students or parents doing home projects, like finding the right-sized tiles for a square floor, can use square root tricks to estimate side lengths when only area measurements are given.

Download Worksheets

Problem 1

What is the square root of 144?

Okay, lets begin

 12
 

Explanation

144 is a perfect square.
Try small numbers:
12 × 12 = 144
So, the square root of 144 is ±12.
 

Well explained 👍

Problem 2

What is the square root of 625?

Okay, lets begin

 25
 

Explanation

Group the digits: (6)(25)
The last digit is 5, so the square root will also end in 5.
The first pair lies between 22 = 4 and 32 = 9. 9 is too big so the tens digit is 2.
Combine the digits: 25 × 25 = 625
So, the square root of 625 is ±25.
 

Well explained 👍

Problem 3

What is the square root of 100?

Okay, lets begin

10
 

Explanation

Group the digits: (1)(00)
The last digit is 0, so the square root also ends in 0.
The first pair is 1, 12 = 1, so the tens digit is 1.
Try 10 × 10 = 100
So the square root of 100 is ±10.
 

Well explained 👍

Problem 4

What is the square root of 961?

Okay, lets begin

31
 

Explanation

Group the digits: (9)(61)
The last digit is 1, so the square root may end in 1 or 9 as 12 = 1 and 92 = 81. 
The first pair is 9, 32 is 9. So the ten’s digit is 3.
Try: 31 × 31 = 961
39 × 39 = 1521
So, the square root of 961 is ±31.
 

Well explained 👍

Problem 5

What is the square root of 1024?

Okay, lets begin

32
 

Explanation

Group the digits: (10)(24)
The last digit is 4, therefore the square root ends in 2 or 8
The first pair is 10, which lies between the squares of 3 and 4, that is, between 32 and 42. Therefore, the tens' digit is 3.
Try: 32 × 32 = 1024
So, the square root of 1024 is ±32.
 

Well explained 👍

FAQs on Square Root Tricks

1.Why do we need square root tricks?

Square root tricks help us find square roots easily and quickly without using a calculator or long division method.

2.Can these tricks be used for all numbers?

These tricks work well for perfect square numbers. For non-perfect squares, they give a close approximation, but not the exact value. 
 

3. What are perfect square numbers?

A perfect square is a number that is the square of a whole number.
 

4.Do square root tricks also work for decimal numbers?

Some tricks can help to estimate square roots of decimal numbers, but for finding exact values, use a calculator or long division method.
 

5.What is digit pairing?

Grouping the digits in pairs starting from the right side is known as digit pairing.