Perfect Square Calculator
Test whether a whole number is a perfect square. Enter any non-negative integer — the calculator takes the square root, checks whether it comes out whole, and shows the nearest perfect squares below and above.
Example: with Number to test 144 → Is it a perfect square?: Yes — 144 = 12².
- Square root12 — a whole number, which is exactly the test
- Nearest perfect square below121 = 11²
- Nearest perfect square above169 = 13²
Computed by the calculator below using its default values. Change any input to see your own numbers.
A perfect square is an integer that equals another integer times itself: n = k². The test is simple — take √n and see whether it is a whole number.
What makes a number a perfect square
A perfect square is what you get by multiplying a whole number by itself: 1, 4, 9, 16, 25, and so on. 144 qualifies because 12 × 12 = 144; 200 does not, because √200 ≈ 14.142 falls between 14² = 196 and 15² = 225. The test in this calculator is exactly that: take the square root, round it, and square it back — if you recover the original number, it is a perfect square.
Perfect squares thin out as numbers grow. There are 10 of them up to 100, but only 31 more between 100 and 1,000, because consecutive squares are 2k + 1 apart — the gap between 12² and 13² is 25, while the gap between 100² and 101² is 201.
Quick screens before you compute
You can often rule a number out by its last digit: perfect squares only end in 0, 1, 4, 5, 6, or 9. Anything ending in 2, 3, 7, or 8 is out immediately. Squares ending in 5 always end in 25, and a square can only end in an even number of zeros. Another screen: the digital root of a perfect square is always 1, 4, 7, or 9. These filters reject most non-squares in your head; the square-root test settles the survivors.
How it’s calculated
Test: n is a perfect square when n is a non-negative integer and round(√n)² = n. Neighbors: for √n between whole numbers, floor(√n)² is the perfect square below and ceil(√n)² the one above; for a perfect square k², the neighbors are (k−1)² and (k+1)². Exact for integers up to 2⁵³ (double-precision limit).
Whole numbers only — fractions like 2.25 = 1.5² are squares of rationals but are not called perfect squares, which are defined over the integers.
The first 20 perfect squares
| k | k² |
|---|---|
| 1 – 5 | 1, 4, 9, 16, 25 |
| 6 – 10 | 36, 49, 64, 81, 100 |
| 11 – 15 | 121, 144, 169, 196, 225 |
| 16 – 20 | 256, 289, 324, 361, 400 |
Computed as k² for k = 1 to 20.
Common mistakes
- Calling any number with a nice square root a perfect square — 2.25 has root 1.5, but perfect squares must be whole numbers with whole roots.
- Trusting a calculator display: √152,399,025 shows 12,345 on many screens, but √152,399,026 shows 12,345.0000000405 — always square the rounded root back to confirm.
- Forgetting 0 and 1 both count (0 = 0², 1 = 1²), which trips up loop bounds in programming checks.
- Assuming a negative number can be a perfect square — squares of real numbers are never negative, so −144 fails even though 144 passes.
Frequently asked questions
What is a perfect square?
An integer n that equals k² for some whole number k. The test: √n must come out to a whole number. 144 = 12² is a perfect square; 145 is not, because √145 ≈ 12.042.
Is 1 a perfect square? Is 0?
Yes to both: 1 = 1² and 0 = 0². Zero is the smallest perfect square, so 0 and 1 open the list 0, 1, 4, 9, 16, …
How can I tell a number is not a perfect square without computing?
Check the last digit — squares never end in 2, 3, 7, or 8. A number ending in 5 must end in 25, and trailing zeros must come in pairs. These screens reject most candidates instantly.
Can negative numbers be perfect squares?
No. Squaring any real number gives a non-negative result, so −144 has no whole-number square root. (Its square roots are imaginary: ±12i.)
What are the perfect squares between 100 and 200?
100, 121, 144, 169, and 196 — the squares of 10 through 14. The next one, 15² = 225, overshoots 200.