Perfect Squares - List of Squares and Formula

Q: Is 0 a perfect square?

Yes. $0 = 0^2$. Whether to include it depends on whether the textbook restricts perfect squares to positive integers or allows non-negative integers; most modern definitions include $0$.

Q: How many perfect squares are there?

Infinitely many. For every integer $n$, the number $n^2$ is a perfect square, and there is no largest integer.

Q: Why does a perfect square always have an odd number of divisors?

Because divisors usually pair up — for $12$, the pair $(2, 6)$ multiplies to $12$, and so does $(3, 4)$. Perfect squares have one divisor that pairs with itself ($\sqrt{n}$), so the total count is odd. For $36$: $1, 2, 3, 4, 6, 9, 12, 18, 36$ — nine divisors, with $6$ pairing with itself.

Q: What is the smallest perfect square greater than 1000?

$1024$, since $32^2 = 1024$ and $31^2 = 961$.

A perfect square is a non-negative integer that equals the product of an integer with itself — written as $n^2$ where $n$ is a whole number. The first few perfect squares are $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ , and the list extends without limit. Perfect squares show up everywhere from area calculations and the Pythagorean theorem to the digit-pattern tricks students use to check arithmetic.

Quick Reference

Field	Value
Definition	An integer of the form $n^2$ where $n$ is a non-negative integer
Symbol	$n^2$
First ten perfect squares	$1, 4, 9, 16, 25, 36, 49, 64, 81, 100$
Formula	$\text{Perfect square} = n \times n = n^2$
Type	Set of non-negative integers
Used in	Algebra, geometry (areas), number theory, the Pythagorean theorem

What is a Perfect Square?

A perfect square is the result you get when you multiply a whole number by itself. The number $49$ is a perfect square because $7 \times 7 = 49$ . The number $50$ is not, because no whole number squared lands on $50$ — the square root of $50$ is $\sqrt{50} \approx 7.07$ , not an integer.

Perfect squares are sometimes called square numbers for an old reason. If you arrange that many objects into a grid with equal sides, you get an exact square.

Try it with $9$ pebbles ( $3 \times 3$ ) or $25$ pebbles ( $5 \times 5$ ). It does not work for $10$ or $26$ .

The List of Perfect Squares From 1 to 30

$n$	$n^2$	$n$	$n^2$	$n$	$n^2$
1	1	11	121	21	441
2	4	12	144	22	484
3	9	13	169	23	529
4	16	14	196	24	576
5	25	15	225	25	625
6	36	16	256	26	676
7	49	17	289	27	729
8	64	18	324	28	784
9	81	19	361	29	841
10	100	20	400	30	900

Memorising the squares of $1$ – $20$ will pay dividends in algebra, geometry, and arithmetic for the rest of school.

The Formula And The Digit-Pattern Shortcut

The formula is the simplest in mathematics:

$n^2 = n \times n$

For two-digit numbers ending in $5$ , there is a shortcut worth knowing. To square $35$ : take the tens digit ( $3$ ), multiply it by the next integer up ( $4$ ), giving $12$ . Then write $25$ at the end.

So $35^2 = 1225$ . The same method gives $45^2 = 2025$ , $65^2 = 4225$ , $95^2 = 9025$ .

A useful fact about the units digit: a perfect square always ends in $0, 1, 4, 5, 6,$ or $9$ . If a number ends in $2, 3, 7,$ or $8$ , it cannot be a perfect square. That alone rules out half the candidates instantly.

How To Check Whether A Number Is A Perfect Square

Example 1: Is $196$ a perfect square?

The units digit is $6$ — possible. Take the square root: $\sqrt{196} = 14$ . That is an integer, so yes.

Final answer: $196 = 14^2$ is a perfect square.

Example 2: Is $250$ a perfect square?

The units digit is $0$ — possible. Take the square root: $\sqrt{250} \approx 15.81$ . Not an integer.

Final answer: $250$ is not a perfect square.

Example 3: Find the next perfect square after $144$ .

The square root of $144$ is $12$ . The next integer is $13$ , and $13^2 = 169$ .

Final answer: The next perfect square after $144$ is $169$ .

The Mathematicians Behind Perfect Squares

Perfect squares are among the oldest objects in mathematics. The Babylonian tablet YBC 7289 (around 1800 BCE) carries an approximation of $\sqrt{2}$ accurate to six decimal places — proof that scribes were already working with non-perfect squares and asking what fell between them.

Pythagoras of Samos (c. 570–495 BCE, Greece) built an entire philosophical school around square numbers and triangular numbers. The Pythagorean theorem $a^2 + b^2 = c^2$ is a statement about three perfect squares — when the squares of two legs add to the square of the hypotenuse, the triangle is right-angled.

Pierre de Fermat (1607–1665, France) found a deep result called the two squares theorem: a prime number can be written as the sum of two perfect squares if and only if it is $2$ or leaves remainder $1$ when divided by $4$ . So $5 = 1^2 + 2^2$ and $13 = 2^2 + 3^2$ , but $7$ and $11$ cannot be written that way at all.

Common Confusions

Perfect square vs perfect cube. A perfect square is $n^2$ ; a perfect cube is $n^3$ . The number $64$ is both — it is $8^2$ and $4^3$ — but most numbers are one or the other or neither.

Square root vs perfect square. The square root operation takes a number and asks "what was squared to get this?". A perfect square is the output of squaring. So $9$ is a perfect square; $\sqrt{9} = 3$ is its square root.

Decimal squares. $2.5 \times 2.5 = 6.25$ . Is $6.25$ a perfect square? Not in the standard sense — the term perfect square is reserved for non-negative integers. $6.25$ is the square of a non-integer, which makes it a square but not a perfect square.

Where Perfect Squares Appear

Beyond the obvious — areas of squares, the Pythagorean theorem, the quadratic formula — perfect squares show up in unexpected places. The number of distinct ways to tile certain rectangles, the structure of magic squares, the divisor counts in number theory (only perfect squares have an odd number of divisors), and the digital roots students use to check long arithmetic. The deeper you go, the more they recur.

If your child is comfortable with squaring single-digit numbers, the natural next step is squaring two- and three-digit numbers and using the result to estimate square roots. At Bhanzu, trainers connect each perfect square to its geometric meaning so the formula doesn't sit isolated.

Was this article helpful?

Your feedback helps us write better content

Frequently Asked Questions

Is 0 a perfect square?

Yes.

0 = 0^2

. Whether to include it depends on whether the textbook restricts perfect squares to positive integers or allows non-negative integers; most modern definitions include

0

How many perfect squares are there?

Infinitely many. For every integer

n

, the number

n^2

is a perfect square, and there is no largest integer.

Why does a perfect square always have an odd number of divisors?

Because divisors usually pair up — for

12

, the pair

(2, 6)

multiplies to

12

, and so does

(3, 4)

. Perfect squares have one divisor that pairs with itself (

\sqrt{n}

), so the total count is odd. For

36

1, 2, 3, 4, 6, 9, 12, 18, 36

— nine divisors, with

6

pairing with itself.

Can a negative number be a perfect square?

No. Squaring any real number — positive or negative — gives a non-negative result.

(-3)^2 = 9

, not

-9

What is the smallest perfect square greater than 1000?

1024

, since

32^2 = 1024

and

31^2 = 961

✍️ Written By

Bhanzu Team

Content Creator and Editor

Bhanzu’s editorial team, known as Team Bhanzu, is made up of experienced educators, curriculum experts, content strategists, and fact-checkers dedicated to making math simple and engaging for learners worldwide. Every article and resource is carefully researched, thoughtfully structured, and rigorously reviewed to ensure accuracy, clarity, and real-world relevance. We understand that building strong math foundations can raise questions for students and parents alike. That’s why Team Bhanzu focuses on delivering practical insights, concept-driven explanations, and trustworthy guidance-empowering learners to develop confidence, speed, and a lifelong love for mathematics.