Squaring is the highest-return thing in Vedic Maths for a school student. It turns up in Pythagoras, in areas, in percentages, in standard deviation and in every quantitative aptitude paper ever set. And it is one of the few places where a shortcut is not merely faster but genuinely effortless.
There are three methods worth knowing. Each has a clear trigger, so you are never wondering which to use, and between them they cover almost every number you will meet.
1. Anything ending in 5
This is the one everybody should know, and it takes about a minute to learn.
Take the part before the 5. Multiply it by the next number up. Write 25 on the end.
35² front is 3 → 3 × 4 = 12 → 1225
65² front is 6 → 6 × 7 = 42 → 4225
75² front is 7 → 7 × 8 = 56 → 5625
95² front is 9 → 9 × 10 = 90 → 9025
It works for longer numbers too
The rule never cared how long the front part was. For 105² the front is 10, so 10 × 11 = 110, and the answer is 11025. For 115² the front is 11, so 11 × 12 = 132, giving 13225.
The only thing that gets harder is the multiplication at the front, which is now two digits by two digits. That is exactly the job of vertically and crosswise, which is why these two techniques are worth learning together.
Why it cannot fail
A number ending in 5 is 10a + 5. Square it:
(10a + 5)² = 100a² + 100a + 25 = 100 × a(a + 1) + 25
The first term is a(a+1) shifted two places left, and the 25 sits in the last two. That is the rule. Understanding this once is worth more than drilling it twenty times, because it tells you there is no awkward case waiting.
2. Anything near 100
Find the distance from 100. Apply that distance to the number again, in the same direction. Square the distance for the last two digits.
98² 2 below → 98 − 2 = 96 | 2² = 04 → 9604
94² 6 below → 94 − 6 = 88 | 6² = 36 → 8836
103² 3 above → 103 + 3 = 106 | 3² = 09 → 10609
🔴 The right-hand part gets exactly two digits. This is the only rule people break. When the square is small, pad it: 2² is written 04, not 4. Write 964 for 98² and you are out by a factor of ten.
When the right-hand part overflows
For 112²: the distance is 12 above, so the left is 112 + 12 = 124, and 12² = 144. But 144 needs three digits and only two are available.
So carry. Keep 44 on the right and add the 1 to the left: 124 + 1 = 125. The answer is 12544.
The same trick works around 50
Less well known and very useful, because it covers the middle of the two-digit range where the other methods run out.
Near 50, the left-hand part starts from 25, not from 50. Add the distance if above, subtract if below. The right-hand part is the square of the distance, still in two digits.
- 53²: 3 above → 25 + 3 = 28, and 3² = 09 → 2809
- 48²: 2 below → 25 − 2 = 23, and 2² = 04 → 2304
- 47²: 3 below → 25 − 3 = 22, and 3² = 09 → 2209
The 25 surprises people. It comes straight out of the algebra: (50 + e)² = 2500 + 100e + e², and 2500 is 25 shifted two places. Use 50 instead of 25 and every answer is wrong, which is the single most common mistake with this variant.
3. Everything else: move to a round number
For numbers that end in neither 5 nor sit near a base, there is a method that always works.
Move up to the nearest round number and down by the same amount. Multiply those two. Add the square of the amount you moved.
47² = 50 × 44 + 3² = 2200 + 9 = 2209
62² = 60 × 64 + 2² = 3840 + 4 = 3844
39² = 40 × 38 + 1² = 1520 + 1 = 1521
88² = 90 × 86 + 2² = 7740 + 4 = 7744
The reason is one line of school algebra you already know:
(n + d)(n − d) = n² − d², so n² = (n + d)(n − d) + d²
Choose d to land on a multiple of ten and the multiplication becomes trivial. Notice 47² appears twice on this page, once by the base-50 method and once by this one, and both give 2209. Two different routes to the same answer is a useful thing to show a sceptical student.
Which one, and when
| If the number | Use | Example |
|---|---|---|
| ends in 5 | front × next, then 25 | 75² = 5625 |
| is within about 10 of 100 | base 100 | 98² = 9604 |
| is within about 6 of 50 | base 50, starting from 25 | 53² = 2809 |
| is anything else | move to a round number | 62² = 3844 |
Choosing quickly is most of the speed. Someone who knows one method and forces every number through it will be slower than someone who knows three and picks in half a second.
Check before you commit
Squaring shortcuts fail in a specific way: they give an answer that looks perfectly reasonable and is out by a factor of ten, usually from forgetting to pad the right-hand part to two digits.
Take 94² = 8836. The digit sum of 94 is 4, and 4 × 4 = 16, which reduces to 7. The digit sum of 8836 is 8 + 8 + 3 + 6 = 25, which reduces to 7. They agree.
Five seconds, and it catches the padding error every time, because dropping a digit changes the digit sum. What it cannot catch is worth knowing too: the digit sum check and its blind spots.
Where this is taught
Ms. Ishita Parikh holds International Vedic Maths Teachers' Training certification at Grade A from IIVA, approved under Skill India and the NSDC, along with two teaching diplomas for young children.
Classes are live online in small batches, covering all 16 sutras in an order built around the student rather than the textbook. See the Vedic Maths course, or book a free demo class. If you are teaching a child yourself, the order these are introduced in matters more than most people expect.
Questions people ask
How do you square a number ending in 5?
Take the digits before the 5, multiply that number by the next number up, and write 25 after it. For 35, the front is 3, so 3 times 4 is 12, and the answer is 1225. For 75, 7 times 8 is 56, giving 5625. It works because a number ending in five is 10a plus 5, and squaring that gives 100 times a times a plus one, plus 25, which is exactly the two parts you wrote down. There is no case where it fails.
What is the fastest way to square a two-digit number?
It depends on the number, and picking the right method is most of the speed. If it ends in 5, use the front times the next number up, then 25. If it is close to 100 or close to 50, use the base method, which needs one subtraction and one small square. For anything else, move to the nearest round number and back by the same amount, multiply those two, and add the square of what you moved: 47 squared becomes 50 times 44 plus 9, which is 2209. Between them these three cover nearly every two-digit number.
How do you square numbers near 100?
Find how far the number is from 100, apply that same distance to the number again in the same direction, and write the square of the distance as the last two digits. For 98, the distance is 2 below, so 98 minus 2 is 96, and 2 squared is 04, giving 9604. For 103, the distance is 3 above, so 103 plus 3 is 106, and 3 squared is 09, giving 10609. The right-hand part must occupy exactly two digits, so pad it with a zero when it is small and carry into the left when it runs over.
Does the ending in 5 trick work for three-digit numbers?
Yes, without modification, because the rule never cared how long the front part was. For 105 the front is 10, and 10 times 11 is 110, so the answer is 11025. For 115 the front is 11, and 11 times 12 is 132, giving 13225. The only thing that gets harder is the multiplication at the front, which is now two digits by two digits rather than one by one, and that is exactly the job of the vertically and crosswise method.
Why does 35 squared equal 1225?
Because 35 is 30 plus 5, and squaring that gives 900 plus two lots of 150 plus 25, which is 900 plus 300 plus 25. The first two terms are 1200, or 100 times 3 times 4, and that is where the shortcut comes from: the front digit multiplied by the next number up, carrying a factor of 100, with the 25 left over on the end. Seeing the derivation once is worth more than memorising the rule, because it tells you the rule cannot fail.