# Shortcut to Find Square of a Number

Today I will discuss a very simple method of finding square of numbers between 26 to 74 mentally. In the subsequent post we will cover higher numbers. So keep watching this space to learn squaring any number within your mind

**Square **(also called** perfect square**) is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3.

## How to find the square of any number?

To apply this method you should know squares of 1 to 25 by heart. You can refer to this table to learn the same.

Number |
Square |
Number’ |
Square’ |

1 | 1 | 13 | 169 |

2 | 4 | 14 | 196 |

3 | 9 | 15 | 225 |

4 | 16 | 16 | 256 |

5 | 25 | 17 | 289 |

6 | 36 | 18 | 324 |

7 | 49 | 19 | 361 |

8 | 64 | 20 | 400 |

9 | 81 | 21 | 441 |

10 | 100 | 22 | 484 |

11 | 121 | 23 | 529 |

12 | 144 | 24 | 576 |

* | * | 25 | 625 |

**For finding square of any number between 26 to 75**

Step 1. Find the difference between 50 and the number you want to square.

** Scenario 1:** If the number to be squared is greater than 50

Step 2. Add that many 100s to 2500 (which is the square of 50)

Step 3. Then add the square of the difference to the result of step 1

*Scenario -2**: If the number is less than 50*

Step 2. Subtract that many 100s to 2500.

Step 3. Then add the square of the difference to the result of step 1

**Example**

Find out the Square of 67.

Step 1. Difference of 67 and 50 = 67-50 = 17

Step 2. This number is greater than 50. So add 1700 to 2500 = 4200

Step 3. Add square of 17 to step 2.

**Answer = 4200+ 289 = 4489**

Alternative method of calculating the square of a number:

Since, 67-50 = 17

67^2

We will be getting answer in 2 parts; see below – right hand side gives you tens and units digit. Left hand side gives you the remaining digits.

= **25 + 17** **|** **(17)^2** ( **|** denotes separation )

= **42** **|** ** _{2}89** (17^2 is 289. The 2 shown in subtext will be carried over and added to left hand side)

= 4489

*Check out another shortcut and quick method of finding square of any 2 digit number. *

To find the square of any small number, we can easily do it by using expansions.

For example, if we want to square 47 , we can do it by changing the number into two numbers.

47×47=(50-3)the whole square.

Then,by using expansions, the result is

50square + 3square – 2×50×3

Which is nothing but 2203.

The answer!!!

sir.. the Ans will be 2209 in place of 2203

…

How to solve 178×225 by esy trick?

Make it (178*900)/4. Now 89*9 can be calculated more easily (89 came after cancelling 178/4 by 2 each). If you see closely, 89*9 be calculated as 90*9 and then subtract 9 from there which will come out to be 801. Now divide 80100 by 2 which will be 40050 which again you do not need to CALCULATE but can find out just by looking at numbers. So the whole calculation can be solved without using a pen.

dear sir,

i would like to know the shortcut for multiplication

4489* 67

find shortcut method to find square root between 40 and 50 , 50 and 60 ,90 and 100

For squaring of any 2 digit number.

Example:-

(67)^2=

Get to its nearest 10’s

1st. As 67, its nearest 10 is 70, so 70-67=3

Then 67-3=64,

64*70=4480—1,

2nd. As 70 is 3 more than 67, so

(3)^2=9—2,

Adding 1&2- 4480+9=4489 s here it is..

(67)^2=4489.

a method to find square of a no.

• Consider number 121 to find square.

• Separate digits as 12 & 1.

• Now do Square of unit digit number i.e. 1’square=1, so our unit digit number is 1 now in this case.

• Then multiply both separated Digits which are earlier separated ie.12 & 1

• =12 X 1=12

• Now double the result of multiplication i.e. Double of 12= 24. And place it left after unit digit number i.e. 2 41.( 2 is Carry)

• Separate carry 2 then

• Now At last square 12 i.e. 12’square=144 and add earlier Carry to it.

• So we get 144+2(carry)=146

• Finally we obtained Square as 14641..

Let us have a look at one more example.

151’square:

• 1’s square=1.

• 2 x (15 x 1) = 3 0 (Double of 15 and 1 and 3 is carry now.)

• Now 15’square=225+earlier carry i.e.3=228

• So Final Answer is 22801

here’s another way

Let us first find square of 11 using formula:

11’square=11+1/1square=12/1=121.

The formula is self explanatory. However, let me explain it in detail for more clarification.

• The slash is used just as a operator.

• Our operating zone is 10 X 1 or simply 10.

• 11 is more than 10.

• We add 1 to 11to make 12.

• The number of digits after the slash can be only one.

• If the number of digits after the slash exceeds one, then we place only the rightmost digit on the extreme right after the slash, and the remaining gets added to the number on the left hand side of the slash.

• Now have a look at few more examples for better understanding.

12’square=12+2/2’square=14/4=144

13’square=13+3/3’square=16/9=169.

14’square=14+4/4’square=18/16 (Apply step no 6 here) 18/16=18+1/6=196.

15’square=15+5/5’square=20/25=20+2/5=225.

16’square=16+6/6’square=22/36=22+3/6=256.

You can work like this up to 19’square.But for 20 formula is slightly change.

The slight Change in formula as follows:

21’square=2 X (21+1)/1’square= 2 X (22)/1=44/1=441.

This change is because now we are operating in the 10 X 2 Zone. Similarly we can calculate Square of 31 but with slight change as follows:

31’Square=3 X (31+1)/!’square= 3 X (32)/1=96/1=961.

By these methods explained you can easily calculate and memorize the squares of numbers up to 99 with out much hassle.

Master Formula to Calculate Square in 10 Seconds.:

Till now we have seen various formulae to calculate square of the number. Now I am giving such a master Formula by which you can orally calculate Square of the number with in 10 Seconds.

11’square=121. 12’square=144

111’square=12321 121’square=14641

Thank you very much for the help.

this trick is only applicable for 2-digit number

suppose we have 65 then

step1: look for digits i.e. 2 then multiply it with 2 then 2*2=4

step2: take 4 places _ _ _ _

step3:square both no.s 6*6=36 and 5*5=25

step4: put the values in sequence i.e. 3625

step5:multiply 6 and 5 and then with 2 i.e. 6*5*2=60

step6:leaving one space add to 3625 i.e.

3625

+60

——

4225

answer=4225

Superb trick. Thank you so much

Suppose the number is 55 then multiply ten’s place digit with the previous number i.e., 5(4*5), we get 20 and the last digit is 25, by default. So we get the answer is 2025

Superb yar seriously …………….

how to calculate square of 3 digits

Trick for squares ends with 5 number.. Take 65. Now seperate like 2 numbers 6/5

step1: square of 5 is 25

step2: 6 immediate number is 7. 6 multiply with 7. Ans is 42.

Step3:42/25.. Ans is 4225

For sqaure root:-

(for all non perfect sqaures)

Find two perfect squares that your square falls between. For example if I am trying to find the square root of 12, then I know my number is going to fall between the square root of 9 (3^2=9) and the square root of 16 (4^2=16).

Divide your square by one of these two square roots. Therefore, I am going to divide my square, 12, by one of the square roots 3 or 4. I will choose 3. So, 12/3 = 4.

I will average the result from Step 2 with the root I divided by. So, I take my answer from Step 2 (4), and will average this with the root I chose to divide my square by in Step 2 (3). Therefore, (4+3)/2 = 3.5. The square root of 12 is 3.5!

try this with large numbers. for very small numbers like 2, 3 etc it doesn’t work i.e. as you try to find the square root of larger numbers you will get more precise answer……

I know one easiest way to find the square root.

Ex: I want to find out the square of 76

Explanation :

a) 76 is nearer to 80 than 70

b) square of 80 is 6400 we know that ( 8* 8 = 64*100= 6400)

c)know multiply the 80 with twice difference of 80 and 76 (80-76 =4) (4*2 =8)

d) 8*80 = 640

e) subtract 640 from 6400 and add 6 (it is the last digit of 6 sqare) at ones digit. you got the answer

Exception=

1) if the number ends with 4 we need to add 16 or ends with 5 we need to add 25

if we go with the 70 and 76 then simply add(76-70) 6*2*70+4900+36

For sqaure root:-

(for all non perfect sqaures)

Find two perfect squares that your square falls between. For example if I am trying to find the square root of 12, then I know my number is going to fall between the square root of 9 (3^2=9) and the square root of 16 (4^2=16).

Divide your square by one of these two square roots. Therefore, I am going to divide my square, 12, by one of the square roots 3 or 4. I will choose 3. So, 12/3 = 4.

I will average the result from Step 2 with the root I divided by. So, I take my answer from Step 2 (4), and will average this with the root I chose to divide my square by in Step 2 (3). Therefore, (4+3)/2 = 3.5. The square root of 12 is 3.5!

try this with large numbers. for very small numbers like 2, 3 etc it doesn’t work i.e. as you try to find the square root of larger numbers you will get more precise answer……

sir,

how can i find the square root of a number which is more than 4 digit( using short process)

How to find the square root of a number like 2750.25??

Shabeena plz dont ask too difficult questions.

18623787961

just square 5*5 and keep 25 a side and multiply 7*8 =56 then join them i.e 5625 answer

its wery easy to find square of the two digit num ending with 5

for example

hey let me tel abt the suaring of two digit number ending with 5’s

for ex take 25*25

step 1: multiply 5*5=25

step 2:multiply 2*3=6

the answer is 625

first of all wen u see 25*25 nd 35*35 etc wen ever u fing 2 digit numbers ending with 5 you just write 25 atlast and then multiply the first num with the conseqte num as shown above

if it is 45*45

u just write last 25

and multiply 4*5=20

so answer is 2025

really nice tip..very thankful to u..

its good

hi,thank u very much was very much helpfull.

Is there any trick to find square of a 9 series number like 99, 999, 9999 and so on?

yes

Sir i request u.

i m very confused in square form but i rember 1……..to ….30 sq.and qube form plz help me y any tricks find for any no. of squre….

for two digit square we are having only 9 simple formulas by that we can square all the two digit numbers mail me if u need it

ashok.gujjula@gmail.com

can u plzzzzz mail me those 9 simple formulae…plzzzz

can anybody tell me how to find square of 25-50 and more than 100 from the same trick

25-50 square trick with example

for any two numbers ending with 5 we can easyly find the square

if it is 45*45

u just write last 25

and multiply 4*5=20

so answer is 2025

sir pls give the trick for find out the square of greater than hundred

if the no is 105 then

1.take last 2 digit & squre it,write down in rhs,i.e.25

2.get diff bet 100 & 105 i.e. 5

3.add it to original no, 105+5=110,write in lhs

4. answer=11025

How to find the square of no. 123??

i want to know how to find the square of 136469

hei i know many shortcuts .. i’ll help u catch me at karthikthebest007@gmail.com . iam very very interested and eager in maths . bye…….

kk.

.. i’ll mail u.

If you want to share anything special with all the QuickerMaths.com readers, send it to me at vineetpatawari@gmail.com. If useful, I’ll post it on QuickerMaths.com.

plz help me ,tell me shorcuts of probability ,if u have fb account ,then plz contact me

hey contact me on jay_nfs@yahoo.in for probability shortcuts

how to solve suare root of number greater than 75 ??

how to find square root of number <75 ??

nt only <75 for two digit square we are having only 9 simple formulas by that we can square all the two digit numbers

u can mail me if u want

plz give….

HI !

LEARN THE SQURE ROOT OF 1 TO 25

HI,

CUBE ROOT OF 512000

STEP1 :- CLOSE THE LAST THREE NUMBERS AND THE REMAINND NUMBERS =512

SEE THE IN WHICH NUMBER IS THERE IF IT IS IN BETWEEN THEN LEAST NUMBER CUBE

STEP2 :- AND YOU SEE THE LAST NUMBER 0 IS 0 AND IF IT IS 4 THEN YOU SEE THE CUBE ROOT OF 4 IS 64 YOU SEE ONLY LAST NMBER

plz explain it with example.

Not understanding. Give an example plzz

There is another easy method….

eg : 37 ^2

Step I (Divided into three parts)

Write as 3^2 | 2 * 3 *7|7^2

Step II

Leave on carry to the next part so that only one digit is left in 10th and unit’s place

Then we get 13 | 7 |9

please explain Step 2 once again clearly sir!!!

Thanks Gokul for your suggestion. BTW the answer is 1369 and not 1379.

Actually this is an application of the formula: (a+b)^2 = a^2 + 2*a*b + b^2; here a = 30 and b = 7.

Arranging the same thing vertically will make it easier for you to understand –

900

420

49

——-

1369

If written in the manner suggested by Gokula (vedic maths way), it becomes simpler.

sir u all r doing a great job

thanx Q so much 4 providing such a marvelous trick of finding a square….thank u so much sir…

ya gud trick it works

What is the best way to start planting a flower garden?

if u wanna multiply 2digit number wid D another 2 digit number i hve 1 simple trick wic i wanna share wid U all

e.g

31 X 41

den 1st a fall multiply 1st digit of one number wid D 1st digit of another number (like 3 X 4)

u will get D first 2 digit of ur ans. i.e 12

12 _ _

den multiply 2nd digit of 1st number wid D 2nd digit of another number (like 1X 1)

den u will get D last digit if ur ans i.e 1

12_ 1

know 1 middle digit is remaining

4 this u hve 2 do RAINBOW multiplication

means u hve 2 find D product of outside digit & den inside digit

after this add the product of inside digit and outside digit [ 3 X 1(outside digit) + 4 X 1(inside digit) ]

here u get D result 7

1271 is D ans

by this method u can easily find out D square also

23*45 pls solve vid dis method

sir u all r doing a great job

thanx Q so much 4 providing such a marvelous trick of finding a square

Hi, afcourse this method is very helpfull for finding the squre and squre root

103*105=107,added the last term of 105 of value 5

103+5=107

104*104=108

104+4=108

thanx

Thanks Nandeesh for suggesting further simplification. I am glad we have regular contributors like you.

I think that for both the scenarios of whether number is less than 50 or more than 50, same method as scenario 2 will work.

EX: 67

67 – 25 | (17)^2 ( | denotes separation )

= 42 | 289 (17^2 is 289. The 2 shown in subtext will be carried over and added to left hand side)

= 4489

EX: 37

37 – 25 | (13)^2 ( | denotes separation )

= 12 | 169 (13^2 is 169. The 1 shown in subtext will be carried over and added to left hand side)

= 1369

Hello Dear Nandeesh,

The cases you have mentioned are very specific – I have discussed all of them in my earlier posts

Squaring Number ending in 5 – http://www.quickermaths.com/squaring-number-ending-in-5/

Multiplying 2 numbers, sum of whose units digit is 10 – http://www.quickermaths.com/vedic-multiplication/

I will not say there aren’t simpler methods of finding square than the one mentioned in this post. But this method can be used for large variety of case.

In any case, I appreciate your efforts of describing various methods. I will be happy to post any method suggested by you on QM (if it’s not already published on QM ). I must say your earlier posts were superb.

With due respect to all, I feel, there are shorter methods for finding squares of numbers without need to remember squares of 1 to 25.

Pl. go through the three tricks below:

1. A quick way to square numbers that end in 5:

75^2 = 5625

75^2 means 75 x 75.

The answer is in two parts: 56 and 25. The last part is always 25.

The first part is the first number, 7, multiplied by the number “one more”, which is 8:

so 7 x 8 = 56

Similarly 852 = 7225 because 8 x 9 = 72.

Before trying to square numbers which do not end in 5, please learn the following trick.

2. Method for multiplying numbers where the first figures are the same and the last figures add up to 10.

32 x 38 = 1216

Both numbers here start with 3 and the last figures (2 and 8) add up to 10.

So we just multiply 3 by 4 (the next number up) to get 12 for the first part of the answer.

And we multiply the last figures: 2 x 8 = 16 to get the last part of the answer.

And 81 x 89 = 7209

We put 09 since we need two figures as in all the other examples.

3. Method for squaring numbers that do not end in 5.

Ex: 83 x 83 = 83 x (87 – 4) = 83 x 87 – 83 x 4

As explained in trick 2 above, 83 x 87 = 7221.

83 x 4 = 332.

83 x 83 = 7221 – 332 = 6889

@Nikhil: Ofcourse you can use the same method to find the square of numbers greater than 74. But in my next method I will give you the extension of this technique to find squares of number between 76 to 124 and so on.

Hi , this method seems correct, but if I try to find square of say 89 then from your method i need to know (89-50)= 39^2. so in that case we may have to use to this method twice, at first to find 39^2 and then 89^2