Perfect squares chart
A perfect square is a whole number times itself: 7 × 7 = 49, so 49 is a perfect square and 7 is its square root. This chart lists the squares of 1 to 30, from 1 to 900, with the square root of each.
Free printable PDF, 5th grade to 8th grade.
One page, black and white, prints the same every time.
What is on the sheet
1 to 15
| Square | Square root |
|---|---|
| 1² = 1 | √1 = 1 |
| 2² = 4 | √4 = 2 |
| 3² = 9 | √9 = 3 |
| 4² = 16 | √16 = 4 |
| 5² = 25 | √25 = 5 |
| 6² = 36 | √36 = 6 |
| 7² = 49 | √49 = 7 |
| 8² = 64 | √64 = 8 |
| 9² = 81 | √81 = 9 |
| 10² = 100 | √100 = 10 |
| 11² = 121 | √121 = 11 |
| 12² = 144 | √144 = 12 |
| 13² = 169 | √169 = 13 |
| 14² = 196 | √196 = 14 |
| 15² = 225 | √225 = 15 |
16 to 30
| Square | Square root |
|---|---|
| 16² = 256 | √256 = 16 |
| 17² = 289 | √289 = 17 |
| 18² = 324 | √324 = 18 |
| 19² = 361 | √361 = 19 |
| 20² = 400 | √400 = 20 |
| 21² = 441 | √441 = 21 |
| 22² = 484 | √484 = 22 |
| 23² = 529 | √529 = 23 |
| 24² = 576 | √576 = 24 |
| 25² = 625 | √625 = 25 |
| 26² = 676 | √676 = 26 |
| 27² = 729 | √729 = 27 |
| 28² = 784 | √784 = 28 |
| 29² = 841 | √841 = 29 |
| 30² = 900 | √900 = 30 |
Patterns in the squares
The gap between one square and the next grows by 2 each time: 1, 4, 9, 16, 25 are 3, 5, 7 and 9 apart. A child who knows that can rebuild the list without multiplying. The last digit of a perfect square is always 0, 1, 4, 5, 6 or 9, so a number ending in 2, 3, 7 or 8 is never a perfect square.
The squares of 1 to 12 are the diagonal of the multiplication chart. Past 12 they have to be learned or worked out, and 15² = 225, 20² = 400 and 25² = 625 are the ones worth knowing by heart.
Estimating a square root
A number that is not on the chart sits between two that are. 50 is between 49 and 64, so √50 is between 7 and 8, a little more than 7. That estimate is an 8th grade standard (CCSS 8.NS.A.2), and square roots of perfect squares come in the same year (8.EE.A.2). Sixth grade meets the squares as exponents.
Common questions
- Is 0 a perfect square?
- Yes, because 0 × 0 = 0. The chart starts at 1 because that is where most classroom lists start.
- What is the difference between a square number and a perfect square?
- Nothing. Both names mean a whole number multiplied by itself. "Square number" is the more common name in elementary school.