Charts
Perfect Squares
A perfect square is a whole number multiplied by itself: 1, 4, 9, 16, 25 and so on. This page lists every perfect square from 1² = 1 to 100² = 10,000 in one chart, then shows four quick tests for spotting a square number, the patterns hiding in the list, and the mistakes people make when they check one.
What is a perfect square?
Perfect squares, also called square numbers, are the results of multiplying a whole number by itself. 7 × 7 = 49, so 49 is a perfect square, and 7 is its square root. Arrange 49 counters in 7 rows of 7 and they make an exact square, which is where the name comes from. Try the same with 50 and one counter is always left over.
The first fifteen are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 and 225. The squares from 1 to 144 also run down the diagonal of the multiplication chart, where each row number meets the same column number.
Zero is a perfect square too, since 0 × 0 = 0, though most lists start at 1, the first of the counting numbers described in the guide to whole and natural numbers. Negative numbers are never perfect squares, because a negative times a negative is positive. Squaring is the same as raising to the power 2, and the exponent rules cover what happens when you combine squares with other powers.
Perfect squares chart from 1 to 10,000
Every perfect square from 1² to 100². Read n in the left column of each pair and n² beside it.
| n | n² | n | n² | n | n² | n | n² |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 26 | 676 | 51 | 2,601 | 76 | 5,776 |
| 2 | 4 | 27 | 729 | 52 | 2,704 | 77 | 5,929 |
| 3 | 9 | 28 | 784 | 53 | 2,809 | 78 | 6,084 |
| 4 | 16 | 29 | 841 | 54 | 2,916 | 79 | 6,241 |
| 5 | 25 | 30 | 900 | 55 | 3,025 | 80 | 6,400 |
| 6 | 36 | 31 | 961 | 56 | 3,136 | 81 | 6,561 |
| 7 | 49 | 32 | 1,024 | 57 | 3,249 | 82 | 6,724 |
| 8 | 64 | 33 | 1,089 | 58 | 3,364 | 83 | 6,889 |
| 9 | 81 | 34 | 1,156 | 59 | 3,481 | 84 | 7,056 |
| 10 | 100 | 35 | 1,225 | 60 | 3,600 | 85 | 7,225 |
| 11 | 121 | 36 | 1,296 | 61 | 3,721 | 86 | 7,396 |
| 12 | 144 | 37 | 1,369 | 62 | 3,844 | 87 | 7,569 |
| 13 | 169 | 38 | 1,444 | 63 | 3,969 | 88 | 7,744 |
| 14 | 196 | 39 | 1,521 | 64 | 4,096 | 89 | 7,921 |
| 15 | 225 | 40 | 1,600 | 65 | 4,225 | 90 | 8,100 |
| 16 | 256 | 41 | 1,681 | 66 | 4,356 | 91 | 8,281 |
| 17 | 289 | 42 | 1,764 | 67 | 4,489 | 92 | 8,464 |
| 18 | 324 | 43 | 1,849 | 68 | 4,624 | 93 | 8,649 |
| 19 | 361 | 44 | 1,936 | 69 | 4,761 | 94 | 8,836 |
| 20 | 400 | 45 | 2,025 | 70 | 4,900 | 95 | 9,025 |
| 21 | 441 | 46 | 2,116 | 71 | 5,041 | 96 | 9,216 |
| 22 | 484 | 47 | 2,209 | 72 | 5,184 | 97 | 9,409 |
| 23 | 529 | 48 | 2,304 | 73 | 5,329 | 98 | 9,604 |
| 24 | 576 | 49 | 2,401 | 74 | 5,476 | 99 | 9,801 |
| 25 | 625 | 50 | 2,500 | 75 | 5,625 | 100 | 10,000 |
To check a number that is not in the chart, or to get the square root of a number that is not a perfect square, use the square root calculator. It says whether the number is a perfect square and names the two squares on either side.
How to tell if a number is a perfect square
Four tests, from quickest to most certain. The first two can only rule a number out; the last two give a definite answer.
1. Check the last digit
A perfect square can only end in 0, 1, 4, 5, 6 or 9. Any whole number ending in 2, 3, 7 or 8 is not a square, whatever its size. The reason is that the last digit of n² depends only on the last digit of n:
| n ends in | n² ends in | Example |
|---|---|---|
| 0 | 0 | 20² = 400 |
| 1 or 9 | 1 | 11² = 121, 19² = 361 |
| 2 or 8 | 4 | 12² = 144, 18² = 324 |
| 3 or 7 | 9 | 13² = 169, 17² = 289 |
| 4 or 6 | 6 | 14² = 196, 16² = 256 |
| 5 | 5 | 15² = 225 |
Two refinements catch more fakes. A square ending in 5 always ends in 25 (225, 625, 5625). A square ending in 0 always ends in an even number of zeros, so 1,000 and 90 are out while 3,600 and 4,900 pass.
2. Check the digital root
Add the digits, and keep adding until one digit is left. For a perfect square the result is always 1, 4, 7 or 9. For 1,764: 1 + 7 + 6 + 4 = 18, then 1 + 8 = 9, so it passes. A number with a digital root of 2, 3, 5, 6 or 8 is never a square.
3. Take the square root
If the square root is a whole number, the number is a perfect square. √1,764 = 42, so 1,764 is 42². For 2,000 the root is about 44.72, so it is not; it sits between 44² = 1,936 and 45² = 2,025.
4. Look at the prime factors
In a perfect square, every prime appears an even number of times. That is the definitive test, and it works for numbers far too large to check by eye.
The prime factorization calculator gives the exponents directly. If only one prime has an odd exponent, multiplying by that prime once makes a perfect square: 5,292 × 3 = 15,876 = 126².
Patterns in square numbers
Each square is the sum of odd numbers
Add consecutive odd numbers from 1 and you get the squares in order: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. The first n odd numbers always add to n². The first ten, 1 through 19, add to 100.
The gap between squares grows by 2
The difference between neighboring squares is the next odd number: (n + 1)² − n² = 2n + 1. From 36 to 49 the gap is 2 × 6 + 1 = 13. That is the same fact as the odd-number sum, seen from the other side, and it follows from multiplying out (n + 1)² with the distributive property.
Squaring numbers that end in 5
Take the digits before the 5, multiply them by the next whole number up, and write 25 after the result.
Squaring numbers near 50 or 100
Write the number as a round number plus or minus a small one and expand. 48² = (50 − 2)² = 2,500 − 200 + 4 = 2,304. 97² = (100 − 3)² = 10,000 − 600 + 9 = 9,409. The pattern is (a − b)² = a² − 2ab + b².
Squares have an odd number of factors
Factors usually come in pairs, like 3 × 12 for 36. In a perfect square one pair is a number with itself, 6 × 6, so it only counts once. 36 has nine factors: 1, 2, 3, 4, 6, 9, 12, 18 and 36. The factor calculator lists them for any number. The square of a prime has exactly three factors, such as 1, 7 and 49, which makes it one of the simplest composite numbers.
How many perfect squares are there up to a number?
Count up to N by taking the square root of N and dropping the decimal part. There are 31 perfect squares from 1 to 1,000 because √1,000 ≈ 31.62: 31² = 961 is the last one in range and 32² = 1,024 is past it.
| Range | Number of perfect squares | Largest one |
|---|---|---|
| 1 to 10 | 3 | 9 = 3² |
| 1 to 100 | 10 | 100 = 10² |
| 1 to 1,000 | 31 | 961 = 31² |
| 1 to 10,000 | 100 | 10,000 = 100² |
| 1 to 1,000,000 | 1,000 | 1,000,000 = 1,000² |
The squares thin out as numbers grow. Ten of the first hundred whole numbers are squares, but only a hundred of the first ten thousand.
Squares of fractions and decimals
Strictly, “perfect square” means the square of a whole number. The same idea extends to fractions whose top and bottom are both perfect squares: 9/16 = (3/4)² and 25/49 = (5/7)². Decimals follow along: 0.25 = 0.5² and 1.44 = 1.2².
Numbers that are not perfect squares have square roots that never end or repeat. √2 ≈ 1.41421356 is the best-known, and the square root of 2 guide explains why no fraction can equal it. That is also why the simplest radical form of a root like √72 works by pulling out the largest perfect square factor: √72 = √36 × √2 = 6√2.
Common mistakes
- Doubling instead of squaring. 6² is
6 × 6 = 36, not6 × 2 = 12. - Trusting the last digit alone. Ending in 1, 4, 5, 6 or 9 does not make a number a square: 11, 24 and 99 all pass the last-digit test and fail the square root test.
- Calling a number a perfect square because it is even, or a multiple of 4. 8, 12 and 20 are multiples of 4 and are not squares.
- Thinking 2 or 3 is a perfect square because it has a square root. Every positive number has a square root; a perfect square is one whose root is a whole number.
- Forgetting 1 and 0.
1 × 1 = 1and0 × 0 = 0, so both qualify. - Squaring a sum term by term.
(3 + 4)² = 49, not9 + 16 = 25.
Frequently asked questions
What are the perfect squares from 1 to 100?
There are ten: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. They are the squares of the whole numbers 1 through 10. The gaps between them are the odd numbers 3, 5, 7, 9, 11, 13, 15, 17 and 19, which is a quick way to rebuild the list from memory.
Is 1,764 a perfect square?
Yes. 42 × 42 = 1,764, so 1,764 is 42². Its prime factorization is 2² × 3² × 7², with every exponent even, which confirms it. The last digit 4 and the digital root 9 are both allowed for squares, so the quick tests agree as well.
Is 2,000 a perfect square?
No. The square root of 2,000 is about 44.72, not a whole number. It lies between 44² = 1,936 and 45² = 2,025. You can also tell from the end: 2,000 ends in three zeros, and a perfect square ending in zeros always has an even number of them.
Is 0 a perfect square?
Yes. 0 × 0 = 0, so zero is the square of a whole number and meets the definition. Many charts start at 1 because the squares of the counting numbers are the ones people usually want, but in algebra and number theory 0 is counted as a perfect square.
What is the difference between a perfect square and a square number?
There is none. Both names mean a whole number multiplied by itself, such as 25 or 144. “Square number” is more common in elementary math, where the numbers are drawn as square arrays of dots, while “perfect square” is more common in algebra, where it separates 49 from numbers like 50 that only have irrational square roots.
How many perfect squares are there between 1 and 1,000?
There are 31. The largest is 31² = 961, and the next one, 32² = 1,024, is above 1,000. A quick way to count the perfect squares up to any number N is to take the square root of N and ignore the decimal part: √1,000 ≈ 31.62 gives 31.
Can a perfect square end in 2, 3, 7 or 8?
No. The last digit of a square depends only on the last digit of the number being squared, and the ten possible endings 0 through 9 produce only 0, 1, 4, 5, 6 and 9. So any whole number ending in 2, 3, 7 or 8, such as 1,522 or 98, cannot be a perfect square.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/perfect-squares/