Perfect Square Calculator

A perfect square is an integer that can be expressed as the product of two equal integers. Since $12 \times 12 = 144$ is an integer, 144 is a perfect square and its square root, 12, is also an integer.

If you enter a number in the upper field this calculator will instantly show if it is a perfect square or not. If it is not a perfect square, then the square root and the two closest perfect squares are also shown. In list mode you can display a list of all perfect squares up to that sum.

What is a perfect square?

A perfect square is sometimes called a square number. It is the product of two equal integers. If $n$ is an integer, then $n^2$ is a perfect square. Conversely, a number is a perfect square if its square root is an integer.

In this case "integer" means all types of whole numbers including negative numbers, zero and positive numbers but not fractions or decimals. So 0, 1, 4, 9 and 16 are perfect squares while 2, 2.25 and 12.5 are not.

Since squaring a negative number results in a positive number, then $(-5)^2 = 25$ is equal to $5^2 = 25$. Therefore all perfect squares are greater than or equal to zero. Negative numbers can never be perfect squares and if you enter a negative number this calculator will indicate that.

How to determine whether a number is a perfect square.

There are three quick checks to perform and this tool will do them all.

To calculate a square root:

The most direct method of checking is to calculate the square root and check if the result is an integer. As the result of $\sqrt{49} = 7$ is an integer, 49 is a perfect square. On the other hand, $\sqrt{53} \approx 7.28$ is not an integer, so 53 is not a perfect square. These are the values shown for the square root on the panel.

Check last digits:

The last digit of a perfect square is always either 0, 1, 4, 5, 6 or 9. The last digit can never be 2, 3, 7 or 8. So if the last digit of a number is one of those four digits, you can immediately rule it out without doing any calculations.

It is possible to rule out numbers based on the last digit but it cannot be guaranteed that they are a square number. The last digit of 63 is 3 so it is not a square number while the last digit of 62 is also 2 unlike the last digit of 64 which is 4 and therefore the square of 8. A condition for the last digit may be met but this only means that it is worth investigating further.

Prime factorization

One method of checking whether a number is a perfect square is to find its prime factors. If every prime factor appears an even number of times, then the number is also a perfect square. For example, 144 = $144 = 2^4 \times 3^2$, and both exponents 4 and 2 are even, so 144 is a perfect square. However, 150 = $150 = 2 \times 3 \times 5^2$ has some odd exponents, so it's not a perfect square. The factor table in this article provides this information for any number you enter.

List of squares

It is helpful to memorize a few small squares. Here are the first twenty, ranging from 1 to 400.

If you switch the calculator to list mode, you can expand the list as needed and display up to the first hundred numbers. The sum is also calculated with a closed formula for the sum of the first $N$ squares: $1^2 + 2^2 + \cdots + N^2 = \tfrac{N(N+1)(2N+1)}{6}$.

Hidden rules in perfect square numbers

The difference between two consecutive perfect squares is always an odd number, and these odd numbers get larger as they go. The differences between the $1, 4, 9, 16$ are 3, 5, and 7. This means that a perfect square is equal to the sum of a series of consecutive odd numbers: $1 + 3 + 5 + 7 = 16 = 4^2$.

The parity (odd or even) of a perfect square is the same as the number that was squared. The square of an even number is even, and the square of an odd number is odd. All even perfect squares are divisible by 4, because an even number contains a factor of 2, and multiplying it by itself produces two such factors.

An odd perfect square is of the form $4k + 1$. This can be used to make a practical deduction: numbers of the form $4k + 3$ or $4k + 2$ can be immediately eliminated as not being perfect squares. 843 (being $4 \times 210 + 3$) is also not a perfect square.

Where do perfect squares occur?

The term derives from geometry. If you arrange $n^2$ unit squares, you get a square with sides of length $n$ and $n$. This explains why the area of a square with side length $n$ is equal to $n^2$. When an area is a perfect square number, its side lengths are whole numbers.

Square numbers play a central role in the Pythagorean theorem. The sum of the squares of two shorter sides of a right triangle is equal to the square of the longest side. A triple of whole numbers, such as 3, 4 and 5, satisfies this condition because the sum of two square numbers, namely 9 and 16, is equal to 25. Square numbers also appear in quadratic equations, completing the square, and number theory.

Squares and Square Roots

Finding a square and finding a square root are inverse operations. So a perfect square is a number whose square root is an integer. 225 is a perfect square, so $\sqrt{225} = 15$ is a perfect square. On the other hand, $\sqrt{224}$ continues infinitely as an irrational number.

Thus checking whether a number is a perfect square basically consists of checking if its square root is an integer. If you want to calculate the square root itself or need negative numbers with their complex square roots use a tool for calculating square roots. This tool is suitable only if you just want to know if a number is contained in the list of perfect squares.