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 | 289 (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

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Vineet Patawari

Hi, I'm Vineet Patawari. I fell in love with numbers after being scared of them for quite some time. Now, I'm here to make you feel comfortable with numbers and help you get rid of Math Phobia!

70 thoughts to “Shortcut to Find Square of a Number”

  1. 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!!!

    1. 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.

  2. 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.

  3. 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

  4. 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

      1. 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

    1. 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

  5. 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……

    1. 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

  6. 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……

  7. 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

  8. 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….

    1. 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

    1. 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

    1. 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

  9. 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

  10. 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

    1. 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.

  11. 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…

  12. 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

  13. 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

  14. 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.

  15. 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

  16. @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.

  17. 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

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