The square of a number?
4 by 4 grid in outline form
4 by 4 grid in outline form

16 units arranged 4 by 4
16 units arranged 4 by 4

Squared Numbers

The examples below show how squares can be found.

Number
Multiplied by itself
Square
1
1 x 1
1
2
2 x 2
4
3
3 x 3
9
4
4 x 4
16
5
5 x 5
25
6
6 x 6
36
7
7 x 7
49
8
8 x 8
64
9
9 x 9
81
10
10 x 10
100
11
11 x 11
121
12
12 x 12
144
13
13 x 13
169
14
14 x 14
196
15
15 x 15
225

 12 x 12 multiplication grid with squares highlighted
12 x 12 multiplication grid with squares highlighted

Squared Words!

You'll hear different words used when people talk about squares. e.g.
  • Square of a number: 25 is the square of 5
  • Squaring a number: multiplying the number by itself
  • A squared number: 100 is a square number
  • 3 squared: 3 squared is 9
  • What's the square of 9? The square of 9 is 81
  • Perfect square: Perfect square is another term for square number

Writing squares

Squares are powers of two.
We write squares usingwe write 32
We could talk about 3 to the power of 2, or the second power of three but we don't usually do so; we say 3 squared. We write 32

Square Roots

You can think of finding square roots as the opposite of finding squares. You find the square root of a number (let's call it number A) by finding the number that, when multiplied by itself produces number A. The examples below show this:

Number
Square Root


Number
Square Root
1 x 1 =
1
1

9 x 9 =
81
9
2 x 2 =
4
2

10 x 10 =
100
10
3 x 3 =
9
3

11 x 11 =
121
11
4 x 4 =
16
4

12 x 12 =
144
12
5 x 5 =
25
5

13 x 13 =
169
13
6 x 6 =
36
6

14 x 14 =
196
14
7 x 7 =
49
7

15 x 15 =
225
15
8 x 8 =
64
8

16 x 16 =
256
16
Note: Think of these examples: 4 x 4 = 16 and - 4 x - 4 = 16.

Positive numbers have two square roots; a positive (called the principal square root) and a negative. Unless you are asked for the negative square root, you can just give the principal square root.

Writing Square Roots

illustrated example of the square root symbol
illustrated example of the square root symbol

Calculating and Estimating Square Roots

Most calculators have a square root button that quickly calculates square roots. There are other ways of calculating square roots but they aren't quick. If you need an approximate value for a square root you can use a method like the one below.
What is the square root of 42?
What two squares does 42 come between?
36 and 49
What are the square roots of 36 and 49
6 and 7
So the square root of 42 is between 6 and 7.
Let's use some trial and error to get an approximate answer.
Let's try 6.5
42 ÷ 6.5 = 6.46
We're looking to get the number we divide by to be as close to the answer we get as possible. In this case we have 6.5 and 6.46
Close, but we can get closer.
Let's take the average of 6.5 and 6.46 and try that.
42 ÷ 6.48 = 6.48
So we have an square root that is accurate to two decimal places. You can repeat these steps to get as accurate an answer as you want.

Fast math trick to find square root of perfect squares