Math
Perfect Square Calculator
Type any whole number to find out if it is a perfect square.
The calculator returns the exact square root when it is, and otherwise shows the two perfect squares it sits between, how far away each one is, and the decimal square root. Switch modes to list every perfect square between two numbers or to jump straight to the nth perfect square. Everything is computed with exact integer arithmetic, so 40-digit inputs are handled without rounding.
Any whole number up to 60 digits. Commas are fine.
Try an example
Result
Perfect square?
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- Square root
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- Perfect square below
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- Perfect square above
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- Nearest perfect square
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- Prime factorization
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the perfect square calculator solves
A perfect square is a whole number that equals some whole number multiplied by itself: 49 = 7 × 7, 2025 = 45 × 45. This calculator tests any integer, gives the exact root when it exists, and when it does not, tells you which two perfect squares surround it so you can estimate the square root by hand. It also lists perfect squares in a range and finds the nth square.
Three ways to recognise a perfect square
- Take the square root: if √N comes out as a whole number, N is a perfect square. The calculator does this exactly with integer arithmetic, so there is no floating-point doubt for large N.
- Factor it: a number is a perfect square exactly when every A whole number above 1 whose only factors are 1 and itself. in its factorization appears an even number of times. 2025 = 3^4 × 5^2 (even The small raised number saying how many times to multiply the base by itself.), while 50 = 2 × 5^2 has a lone 2.
- Look at the last digit: perfect squares end only in 0, 1, 4, 5, 6, or 9. A number ending in 2, 3, 7, or 8 can never be a perfect square. (Ending in 1, 4, 5, 6, 9, or 0 does not guarantee one.)
| n | n^2 | Gap to next square (2n + 1) |
|---|---|---|
| 1 | 1 | 3 |
| 2 | 4 | 5 |
| 5 | 25 | 11 |
| 10 | 100 | 21 |
| 12 | 144 | 25 |
| 25 | 625 | 51 |
| 45 | 2025 | 91 |
Negative numbers, zero, and decimals
Zero is a perfect square (0 = 0^2), and 1 is the smallest positive one. No negative integer is a perfect square because a real number times itself is never negative; √(-16) is the imaginary number 4i. Decimals like 2.25 = 1.5^2 are squares of rationals but are not perfect squares in the whole-number sense used here, so the checker asks for an integer.
How to use it
- Choose Check one number and type the integer. The result says Yes or No, gives the root, and shows the neighbouring perfect squares.
- Choose List perfect squares in a range and enter the two ends to get every r^2 between them with its root.
- Choose Find the nth perfect square to jump to n^2, the squares before and after it, and the first n squares.
- Open Show the work to see the sandwich estimate, the prime factorization test, and the exact root check.
How to read the answer
Yes means √N is a whole number and the root shown is exact. No means N falls strictly between two perfect squares; the distances tell you which one is closer and the decimal root shows where √N lands between the two integer roots. In list mode the count equals (largest root - smallest root + 1). The prime factorization line explains the answer: all-even exponents means a perfect square.
Common mistakes and edge cases
- Trusting a floating-point square root for big numbers: Math.sqrt(10^30 + 1) prints 10^15 exactly even though 10^30 + 1 is not a perfect square. Use exact integer checks like this page does.
- Assuming a number ending in 5 or 6 must be a square. 15, 26, and 35 are not.
- Forgetting 0 and 1 are perfect squares.
- Calling -25 a perfect square because 25 is; the square of any real number is non-negative.
- Mixing up the perfect square (144) with its root (12) when reading the answer.
Worked examples
Is 2025 a perfect square?
45 × 45 = 2025.
Perfect square?
Yes
Is 50 a perfect square?
Between 49 and 64.
Perfect square?
No
Negative input
No real square is negative.
Perfect square?
No
Zero
0 = 0^2.
Perfect square?
Yes
40-digit perfect square
12345678901234567890^2, checked exactly.
Perfect square?
Yes
Perfect squares between 1 and 200
1, 4, 9, … , 196.
Perfect square?
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196
12th perfect square
12^2 = 144.
Perfect square?
144
Decimal input
Rejected: perfect squares are whole numbers.
Perfect square?
Error
Frequently asked questions
What is a perfect square?+
A whole number that is the square of another whole number: 1, 4, 9, 16, 25, 36, … For example 81 is a perfect square because 9 × 9 = 81.
How do I check if a number is a perfect square without a calculator?+
Estimate the square root, square the nearest whole numbers, and see whether you hit it exactly. For 2025: 40^2 = 1600 and 50^2 = 2500, so try 45: 45^2 = 2025. Alternatively factor the number and check that every prime exponent is even.
Is 0 a perfect square?+
Yes. 0 = 0 × 0. It is usually listed as the first perfect square, with 1 as the first positive one.
Can a negative number be a perfect square?+
No. Any real number squared is zero or positive, so -25 is not a perfect square even though 25 is.
What are the perfect squares up to 100?+
1, 4, 9, 16, 25, 36, 49, 64, 81, 100. There are ten of them, the squares of 1 through 10.
What digits can a perfect square end in?+
Only 0, 1, 4, 5, 6, or 9. A number ending in 2, 3, 7, or 8 is never a perfect square. This is a quick way to rule numbers out, but not to rule them in.
How many perfect squares are there between two numbers?+
Count the whole numbers between the square roots: between 1 and 200 the roots run 1 through 14, so there are 14 perfect squares. The list mode shows them all.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026