Perfect Squares - List of Squares and Formula
Perfect Squares - List of Squares and Formula
Algebra
BT
Last updated on May 10, 2022
A perfect square is a non-negative integer that equals the product of an integer with itself — written as n² 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 n² where n is a non-negative integer |
| Symbol | n² |
| First ten perfect squares | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 |
| Formula | Perfect square = n × n = n² |
| 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 × 7 = 49. The number 50 is not, because no whole number squared lands on 50 — the square root of 50 is approximately 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 × 3) or 25 pebbles (5 × 5). It does not work for 10 or 26.
The List of Perfect Squares From 1 to 30
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 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 |
Memorizing 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:
n² = n × 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 35² = 1225. The same method gives 45² = 2025, 65² = 4225, 95² = 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: √196 = 14. That is an integer, so yes.
Final answer: 196 is a perfect square.
Example 2: Is 250 a perfect square?
The units digit is 0 — possible. Take the square root: √250 is approximately 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 13² = 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 √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 a² + b² = c² 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 a remainder of 1 when divided by 4.
Common Confusions
Perfect square vs perfect cube. A perfect square is n²; a perfect cube is n³. The number 64 is both — it is 8² and 4³ — 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; √9 = 3 is its square root.
Decimal squares. 2.5 × 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.
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, and the digital roots students use to check long arithmetic.
Frequently Asked Questions
Is 0 a perfect square?
Yes. 0 = 0². Whether to include it depends on whether the textbook restricts perfect squares to positive integers; most modern definitions include 0.
How many perfect squares are there?
Infinitely many. For every integer n, the number n² 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 (√n), so the total count is odd.
Can a negative number be a perfect square?
No. Squaring any real number — positive or negative — gives a non-negative result.
What is the smallest perfect square greater than 1000?
1024, since 32² = 1024 and 31² = 961.