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Square Numbers
(KS2, Year 6)
The Lesson
A square number is a number that results from a number being multiplied by itself. For example, 1 is a square number. It is 1 × 1.
 4 is a square number. It is 2 × 2.
 9 is a square number. It is 3 × 3.
Dictionary Definition
The Oxford English Dictionary defines a square number as "the product of a number multiplied by itself."The Square Numbers
The square numbers are:In a number square, the square numbers are shaded below:
These are just the square numbers up to 100. There are infinitely many square numbers, they go on forever.
How to Find Square Numbers
Finding square numbers is easy. Just multiply a number by itself. The first 10 square numbers are found by multiplying 1 by itself, then 2 by itself, as shown below.Interactive Game on Square Numbers
Square Numbers in Algebra
In algebra, letters are used instead of numbers. Just as we can square a number...
2^{2} = 2 × 2 = 4
...we can square a letter:
x^{2} = x × x
Note: We would say x^{2} as "x squared", just like we would say 2^{2} as "2 squared".
Here are some examples of squared letters in algebra:

Quadratic equations have an x^{2} term:

Pythagoras theorem relates the square of the lengths of the sides of a right triangle to each other.
The image below shows the formula for Pythagoras theorem and the lengths of the sides a, b and c of a right triangle:

The area of a circle is related to the square of its radius.
The image below shows the formula for the area of a circle and the circle's radius, r:
Square Numbers Come from Integers
Square numbers are the result of integers (whole numbers, not fractions) being multiplied by themselves. A square number will always be an integer.Square Numbers Are Always Positive
A square number is always a positive number. They can never be a negative number. Negative numbers can be multiplied by themselves to give a positive square number:
−2 × −2 = (−2)^{2} = 4
"Squaring" a Number
The verb which describes the operation of multiplying a number by itself (e.g. 6 × 6) is "to square". When we multiply a number be itself , we call this "squaring" a number. 6 × 6 = 6^{2} (called "6 squared").Why "Square" Numbers?
A square number is a number times itself.
1^{2} = 1 × 1 = 1
2^{2} = 2 × 2 = 4
3^{2} = 3 × 3 = 9
The squares go 1, 4, 9...
A square with sides of length 1 has an area of 1.A square with sides of length 2 has an area of 4.
A square with sides of length 3 has an area of 9.
The areas go 1, 4, 9... The square of a number and the area of a square with side lengths of that number are the same.
Square Roots
If 2² = 4, then 4 is the square of 2. The inverse of the square is the square root, denoted by the √ symbol. 2 is the square root of 4, or √4 = 2. Do you disagree with something on this page?
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