√ Perfect Square Checker
By ToolNimba Math Team · Updated 2026-06-19
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Enter a non-negative whole number to check if it is a perfect square.
A perfect square is a whole number you get by multiplying an integer by itself, so its square root comes out as a whole number with nothing left over. This checker takes any non-negative whole number and tells you straight away whether it is a perfect square. If it is, you see the exact square root. If it is not, you see the nearest perfect squares just below and just above, plus a bonus note on whether the number is also a perfect cube.
What is the Perfect Square Checker?
A perfect square is the product of an integer multiplied by itself. So 1, 4, 9, 16, 25 and 36 are perfect squares because they equal 1×1, 2×2, 3×3, 4×4, 5×5 and 6×6. The defining test is simple: take the square root of the number, and if that root is a whole number (an integer) with no fractional part, the original number is a perfect square. The number 49 passes because its root is exactly 7, while 50 fails because its root is about 7.071, which is not whole.
There are quick ways to rule a number out without a calculator. A perfect square can only end in the digits 0, 1, 4, 5, 6 or 9 in base ten, so any number ending in 2, 3, 7 or 8 is never a perfect square. Perfect squares also leave a remainder of 0 or 1 when divided by 4, and the digit sum of a perfect square reduces to 1, 4, 7 or 9. These shortcuts narrow the field, but the only fully reliable test is to check whether the integer square root, when squared, returns the original number.
This tool checks exactly that. It computes the integer square root, squares it, and compares it to your input, which means the answer is exact even for very large numbers where a floating-point square root would round and mislead. When a number is not a perfect square, knowing the nearest perfect squares on either side is useful for estimating roots by hand and for understanding how far the value sits from a clean root. The bonus cube check applies the same idea to third powers, flagging numbers like 64 that happen to be both a perfect square (8²) and a perfect cube (4³).
When to use it
- Checking quickly whether a number from a homework problem or test is a perfect square before simplifying a square root.
- Simplifying radicals by spotting the largest perfect-square factor inside a square root.
- Verifying that a count of items can be arranged into a perfect square grid (for tiling, seating or pixel layouts).
- Estimating an irrational square root by reading off the nearest perfect squares above and below.
How to use the Perfect Square Checker
- Type a non-negative whole number into the input box.
- Read the Yes or No verdict telling you whether it is a perfect square.
- If it is a perfect square, note the exact square root shown beside the verdict.
- If it is not, use the nearest perfect squares below and above to estimate the root, and check the bonus line to see if it is a perfect cube.
Formula & method
Worked examples
Check whether 144 is a perfect square.
- Take the square root: √144 = 12
- Is 12 a whole number? Yes.
- Confirm by squaring: 12 × 12 = 144, which matches the input.
- Nearest perfect squares: below is 11² = 121, above is 13² = 169.
Result: Yes, 144 is a perfect square with square root 12.
Check whether 150 is a perfect square.
- Find the integer square root: ⌊√150⌋ = 12
- Square it: 12 × 12 = 144, which is not 150.
- Try the next integer: 13 × 13 = 169, which overshoots 150.
- So √150 ≈ 12.247, which is not a whole number.
Result: No, 150 is not a perfect square. It sits between 12² = 144 and 13² = 169.
Check whether 729 is a perfect square (and a perfect cube).
- Integer square root: ⌊√729⌋ = 27
- Square it: 27 × 27 = 729, which matches, so it is a perfect square.
- Cube check: 9 × 9 × 9 = 729, so it is also a perfect cube.
Result: Yes, 729 = 27² and also 9³, so it is both a perfect square and a perfect cube.
Perfect squares from 1 to 15
| Integer n | n squared (n²) |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
Last-digit rule: a perfect square never ends in these digits
| Possible last digit of a perfect square | Last digit that rules it out |
|---|---|
| 0, 1, 4, 5, 6, 9 | 2, 3, 7, 8 |
Common mistakes to avoid
- Trusting a rounded decimal square root. A calculator may show √1000000 as 1000 and √999999 as 1000.0 after rounding, making both look like perfect squares. Only an exact integer check (squaring the whole-number root back) is reliable, especially for large numbers.
- Forgetting that 0 and 1 are perfect squares. Zero is a perfect square because 0 × 0 = 0, and 1 is a perfect square because 1 × 1 = 1. People often skip these edge cases when listing perfect squares.
- Trying to check negative numbers or decimals. Perfect squares are defined for non-negative whole numbers. A negative number has no real square root, and a decimal like 2.25 is the square of 1.5 but is not called a perfect square in the integer sense.
- Confusing a perfect square with a perfect cube. A perfect square comes from an integer times itself (k²), while a perfect cube comes from an integer cubed (k³). A number like 64 is both, but most numbers are at most one of them, so check each separately.
Glossary
- Perfect square
- A whole number equal to an integer multiplied by itself, such as 25 = 5 × 5.
- Square root
- A value that, multiplied by itself, gives the original number. The square root of 36 is 6.
- Integer
- A whole number with no fractional part, such as 0, 1, 2, 3 or larger, with no decimals.
- Integer square root
- The largest whole number whose square is not greater than n, written ⌊√n⌋.
- Perfect cube
- A whole number equal to an integer multiplied by itself three times, such as 27 = 3 × 3 × 3.
Frequently asked questions
What is a perfect square?
A perfect square is a whole number that equals an integer multiplied by itself. For example 49 is a perfect square because 7 × 7 = 49, and its square root, 7, is a whole number. Numbers like 50 are not perfect squares because their square roots are not whole.
How do I know if a number is a perfect square?
Take its square root and check whether the result is a whole number. If the integer square root, when squared, returns the original number exactly, it is a perfect square. This tool does that exact check for you, even for very large numbers.
Is 0 a perfect square?
Yes. Zero is a perfect square because 0 × 0 = 0, so its square root is the whole number 0. The number 1 is also a perfect square, since 1 × 1 = 1.
Can a negative number be a perfect square?
No. Perfect squares are non-negative because any real number multiplied by itself is zero or positive. A negative number has no real square root, so this checker only accepts non-negative whole numbers.
What is the quick last-digit test for perfect squares?
In base ten a perfect square can only end in 0, 1, 4, 5, 6 or 9. So any number ending in 2, 3, 7 or 8 cannot be a perfect square. The test rules numbers out fast, but passing it does not guarantee a perfect square, so a full check is still needed.
Can a number be both a perfect square and a perfect cube?
Yes. A number that is a perfect sixth power is both. For example 64 = 8² and 64 = 4³, and 729 = 27² and 729 = 9³. The tool flags this with a bonus cube check below the main result.