Perfect Square Checker
Enter any whole number to find out instantly whether it is a perfect square. See its square root, the nearest perfect squares above and below, quick mental-math tricks, and a full table of squares.
100% private - runs entirely in your browserWhat Is a Perfect Square?
A perfect square is a whole number that you get by multiplying an integer by itself: 1×1=1, 2×2=4, 3×3=9, 4×4=16, and so on. Geometrically, it is the area of a square whose sides have a whole-number length. Zero also counts, since 0×0=0.
Two-Second Tricks to Rule Numbers Out
1. The last-digit test
A perfect square can only end in 0, 1, 4, 5, 6, or 9. If your number ends in 2, 3, 7, or 8, stop: it is not a perfect square. Example: 1,947 ends in 7, so it cannot be a perfect square.
2. The digital-root test
Add the digits repeatedly until one digit is left. Perfect squares always land on 1, 4, 7, or 9. Example: 169 gives 1+6+9=16, then 1+6=7. But beware: passing this test does not prove anything (145 gives 1 yet is not a square). It only rules candidates out.
Patterns Worth Knowing
- Sum of odd numbers: 1 = 1, 1+3 = 4, 1+3+5 = 9, 1+3+5+7 = 16. The first n odd numbers always add up to n².
- Growing gaps: the gap between n² and (n+1)² is 2n+1. Squares spread further apart as numbers grow: 144 to 169 is 25 apart.
- Prime-factor test: a number is a perfect square exactly when every prime in its factorization appears an even number of times. 144 = 2⁴×3² qualifies; 72 = 2³×3² does not.
- Even/odd: the square of an even number is even, the square of an odd number is odd.
Perfect Squares Table (1-30)
Click any square to check it in the calculator above.
| Root | Square | Root | Square | Root | Square |
|---|---|---|---|---|---|
| 1 | 11 | 21 | |||
| 2 | 12 | 22 | |||
| 3 | 13 | 23 | |||
| 4 | 14 | 24 | |||
| 5 | 15 | 25 | |||
| 6 | 16 | 26 | |||
| 7 | 17 | 27 | |||
| 8 | 18 | 28 | |||
| 9 | 19 | 29 | |||
| 10 | 20 | 30 |
Frequently Asked Questions
What is a perfect square?
A perfect square is a positive integer that equals another integer multiplied by itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. Zero is also a perfect square, because 0 x 0 = 0.
How do you check if a number is a perfect square?
Take the square root. If the result is a whole number with no decimal part, the number is a perfect square. Example: the square root of 144 is exactly 12, so 144 is a perfect square. The square root of 145 is about 12.04, so 145 is not.
Is 144 a perfect square?
Yes. 144 = 12 x 12, so its square root is the whole number 12.
Is 169 a perfect square?
Yes. 169 = 13 x 13, so its square root is the whole number 13.
Is 111 a perfect square?
No. The square root of 111 is about 10.54, which is not a whole number. The nearest perfect squares are 100 (10 squared) below and 121 (11 squared) above.
What are the first 20 perfect squares?
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
Can a perfect square end in 2, 3, 7, or 8?
No. Perfect squares can only end in 0, 1, 4, 5, 6, or 9. If a number ends in 2, 3, 7, or 8, you can rule it out instantly without any more math.
What is the digital root trick for perfect squares?
Add the digits of the number repeatedly until a single digit remains. A perfect square always has a digital root of 1, 4, 7, or 9. Example: 144 gives 1 + 4 + 4 = 9. Note this test can only rule numbers out, it cannot confirm them: 145 has digital root 1 but is not a perfect square.
Why is the sum of odd numbers always a perfect square?
Adding the first n odd numbers always gives n squared: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. This is a classic visual proof you can draw with square tiles.
Is 0 a perfect square?
Yes. 0 = 0 x 0, so zero is the square of zero and counts as a perfect square.
Can a negative number be a perfect square?
No, not among real numbers. Any integer multiplied by itself gives zero or a positive number, so a negative number can never equal an integer squared.
How do prime factors reveal a perfect square?
Factor the number into primes. If every prime appears an even number of times, the number is a perfect square. Example: 144 = 2^4 x 3^2, all exponents even, so 144 is a perfect square. But 72 = 2^3 x 3^2 has an odd exponent, so 72 is not.
What is the nearest perfect square to 50?
49 (7 squared) below and 64 (8 squared) above. The closer one is 49, just 1 away.
How many perfect squares are between 1 and 100?
Ten: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
What is the gap between consecutive perfect squares?
The gap between n squared and (n+1) squared is 2n + 1, which is always the next odd number. Example: from 144 to 169 the gap is 25, and 2 x 12 + 1 = 25.
Where are perfect squares used in real math?
In geometry for the area of squares, in quadratic equations, in number theory and cryptography, and in computer graphics. Recognizing perfect squares quickly also speeds up factoring and mental math.
Free tool by NRTechStudio