Perfect squares are the numbers you get by squaring whole numbers: 1, 4, 9, 16, 25, 36 and onward. They are worth knowing by heart for simplifying radicals, factoring, estimating square roots and spotting patterns. This generator builds a table of squares for any range you choose, or lists every perfect square up to a limit, and shows the gap between each square and the one before it.
How to use the perfect squares list
- Choose a mode: n² for n from … to … lists the squares of a range of whole numbers, and Every perfect square up to a value finds all squares that do not exceed a number.
- Enter the range, for example from 1 to 50, or the largest value, for example 1,000.
- The table appears below with n, n² and the difference from the previous square. The tape shows how many squares were listed, the first and last, and their sum. Up to 2,000 rows can be listed at once, for any n up to one billion.
Formulas behind the table
The second identity means consecutive squares are separated by the odd numbers 1, 3, 5, 7, …, so you can build the whole list by repeated addition. The total of the first n squares is:
Perfect squares from 1 to 30
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 11 | 121 | 21 | 441 |
| 2 | 4 | 12 | 144 | 22 | 484 |
| 3 | 9 | 13 | 169 | 23 | 529 |
| 4 | 16 | 14 | 196 | 24 | 576 |
| 5 | 25 | 15 | 225 | 25 | 625 |
| 6 | 36 | 16 | 256 | 26 | 676 |
| 7 | 49 | 17 | 289 | 27 | 729 |
| 8 | 64 | 18 | 324 | 28 | 784 |
| 9 | 81 | 19 | 361 | 29 | 841 |
| 10 | 100 | 20 | 400 | 30 | 900 |
Worked example: squares up to 200
Choosing “every perfect square up to 200” gives ⌊√200⌋ = 14, so the list runs from 0² to 14²: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196. That is 15 squares, and they add up to 14 × 15 × 29 ÷ 6 = 1,015.
Using the list
Simplifying square roots
To simplify √72, find the largest perfect square that divides 72. Scanning the list: 36 divides 72, so √72 = √36 × √2 = 6√2. The simplify radicals calculator automates this.
Estimating square roots
A square root lies between the roots of the squares around it. Since 49 < 55 < 64, √55 is between 7 and 8, and because 55 is closer to 49, the root is closer to 7 (it is about 7.42).
Spotting non-squares
Perfect squares end only in 0, 1, 4, 5, 6 or 9, and a square ending in 5 always ends in 25. Squares of even numbers are multiples of 4, and squares of odd numbers are always 1 more than a multiple of 8. These checks rule out many numbers instantly.
A note on history
The Greeks pictured square numbers as dots arranged in a square, and noticed that adding the next odd number of dots as an L-shaped border produces the next square. That picture is the identity n² − (n − 1)² = 2n − 1, shown in the difference column of the table.
For a single number, use the square number calculator; to go the other way, use the square root calculator.
Frequently asked questions
What is a perfect square?
A perfect square is the square of a whole number: 0, 1, 4, 9, 16, 25 and so on. Its square root is a whole number, so √49 = 7 exactly while √50 is not a whole number.
How many perfect squares are there below 1,000?
There are 32 if you count 0, from 0² = 0 up to 31² = 961; the next square, 32² = 1,024, is too big. In general the count up to N is ⌊√N⌋ + 1 including zero.
How can I tell if a large number is a perfect square?
Check the last digit first: a perfect square never ends in 2, 3, 7 or 8. Then look at its prime factorization; every prime must appear an even number of times. For example 3,600 = 2⁴ × 3² × 5², so it is 60².
What is the sum of the first n perfect squares?
1² + 2² + … + n² = n(n + 1)(2n + 1) ÷ 6. For n = 50 that is 50 × 51 × 101 ÷ 6 = 42,925, the same total the table shows.