Learn Square Numbers
A square number is produced when a whole number is multiplied by itself, such as 7 × 7 = 49.
Learn the square numbers from 1² to 20², see how perfect squares appear in multiplication tables, and discover patterns that make them easier to recognize.
The notation 7² means 7 multiplied by itself: 7 × 7 = 49.
What Is a Square Number?
A square number is the product obtained when an integer is multiplied by itself. The notation n² means n × n.
If dots or unit tiles are arranged in equal rows and columns, n × n objects form an n-by-n square. For example, 4 × 4 = 16 objects can form a square with four rows and four columns.
Square Numbers From 1 to 12
These twelve perfect squares correspond directly to the multiplication facts on the main diagonal of a 12×12 multiplication chart.
1² = 1, 2² = 4 and 3² = 9.
4² = 16, 5² = 25 and 6² = 36.
7² = 49, 8² = 64 and 9² = 81.
10² = 100, 11² = 121 and 12² = 144.
Square Numbers From 13 to 20
Once the squares through 12² are familiar, you can extend the same idea to larger whole numbers.
13 × 13 = 169
14 × 14 = 196
15 × 15 = 225
16 × 16 = 256
17 × 17 = 289
18 × 18 = 324
19 × 19 = 361
20 × 20 = 400
Where Are Square Numbers on a Multiplication Chart?
Square numbers form the main diagonal of a multiplication chart because the factor at the top and the factor at the side are identical.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144 appear in order along the diagonal of a 12×12 multiplication chart.
Explore the interactive multiplication chart to see these products alongside the other multiplication facts.
Consecutive Square Numbers Differ by Odd Numbers
The gap between one square number and the next is always the next odd number.
For example, 5² = 25. The next odd-number difference is 11, so 25 + 11 = 36, which gives 6² = 36. The next difference is 13, giving 36 + 13 = 49 = 7².
Why Does Adding an Odd Number Make the Next Square?
Imagine building a square from unit tiles. To turn an n-by-n square into the next larger square, you add one new row and one new column around it.
A 3×3 square contains 9 units. Adding 7 more units produces a 4×4 square.
A 4×4 square contains 16 units. Adding 9 more produces a 5×5 square.
Starting from 1, adding successive odd numbers generates the square numbers: 1, 1+3=4, 4+5=9, 9+7=16, and so on.
Square Facts to Know From the Times Tables
The square facts from 1² through 12² are useful multiplication facts because each one belongs to its corresponding times table.
Square Number vs Square Root
Squaring and taking a principal square root move in opposite directions for non-negative numbers.
Start with 8 and multiply it by itself to produce the square number 64.
Start with 64 and ask which non-negative number multiplied by itself produces 64.
12² = 144, while √144 = 12.
Square Numbers Practice
Try to answer each question before opening the solution.
What is 4²? Reveal answer
What is 7²? Reveal answer
What is 9²? Reveal answer
What is 12²? Reveal answer
What is 15²? Reveal answer
What is 20²? Reveal answer
Which square number comes after 144? Use the pattern
Square Numbers Questions
What is a square number?
A square number is the result of multiplying an integer by itself. For example, 6 × 6 = 36, so 36 is a square number.
What are the first 10 positive square numbers?
They are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
What are the square numbers from 1 to 12?
From 1² through 12², the square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.
Is 49 a square number?
Yes. 49 = 7 × 7 = 7².
Is 50 a square number?
No. There is no integer whose square is 50. It lies between 7² = 49 and 8² = 64.
What is 12 squared?
12² = 12 × 12 = 144.
What is 20 squared?
20² = 20 × 20 = 400.
Why are square numbers on the diagonal of a multiplication chart?
Cells on the main diagonal have matching row and column values. They therefore represent products such as 5 × 5, 8 × 8 and 12 × 12, which are square numbers.
What is the difference between a square and a square root?
Squaring multiplies a number by itself, while the principal square root identifies the non-negative number whose square gives the original value. For example, 8² = 64 and √64 = 8.