In mathematics, there are certain operations that we perform repeatedly. One of them is multiplication. When we multiply a number by itself, the result we get is called a square number. The process of multiplying a number by itself is called squaring the number.
The opposite of squaring a number is finding its square root. The square root of a number is the number that, when multiplied by itself, gives that number.
The study of square and square roots is very important in mathematics because it helps us to understand many other topics such as algebra, geometry, mensuration, Pythagoras’ theorem, area of squares, and even real-life calculations.
Before the use of calculators and computers, students used square and square root tables to find values of squares and square roots quickly. These tables are still useful for understanding how these concepts work.
A square number (or a perfect square) is the product of a number multiplied by itself.
This means: Square of a number = number × number
For example:
When we write “2 squared,” we write it as 2².
When we write “5 squared,” we write it as 5².
The small number written at the top right (²) is called a power or index.
So, when a number has ² beside it, it means that number has been multiplied by itself once.
The square root of a number is the value that, when multiplied by itself, gives the original number.
For example:
The symbol used to represent the square root is called the radical sign (√). So, √36 = 6, and 6² = 36.
Squaring and finding the square root are opposite operations. If we square a number, we get a bigger value. If we find its square root, we get back the original smaller value.
A perfect square is a number that is the exact square of a whole number. Its square root is always a whole number (that means it does not have decimal parts).
For example:
That means √1 = 1, √4 = 2, √9 = 3, √16 = 4, and so on.
Perfect squares are very easy to work with because their square roots are neat whole numbers.
Some numbers are not perfect squares. That means their square roots are not whole numbers but decimal or irrational numbers.
For example:
These numbers go on without ending, and they do not repeat in any pattern. That is why they are called irrational numbers.
We can only approximate their square roots (that means write them in short decimal form).
Before calculators were introduced, students and teachers used square tables to find the squares of numbers quickly.
A square table lists numbers and their squares, usually from 1 up to 100 or more.
| Number (n) | Square (n²) |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
You can see that the squares increase very quickly as the numbers increase. The difference between two consecutive squares becomes larger as the number grows.
For example:
4 − 1 = 3,
9 − 4 = 5,
16 − 9 = 7,
25 − 16 = 9,
36 − 25 = 11.
The differences form odd numbers (3, 5, 7, 9, 11…), which shows a pattern in square numbers.
A square root table works in the opposite way of a square table. Instead of showing squares of numbers, it shows square roots of numbers. It helps us find the number that, when multiplied by itself, gives a certain value.
| Number | Square Root (√n) |
|---|---|
| 1 | 1.000 |
| 2 | 1.414 |
| 3 | 1.732 |
| 4 | 2.000 |
| 5 | 2.236 |
| 6 | 2.449 |
| 7 | 2.646 |
| 8 | 2.828 |
| 9 | 3.000 |
| 10 | 3.162 |
| 11 | 3.317 |
| 12 | 3.464 |
| 13 | 3.606 |
| 14 | 3.742 |
| 15 | 3.873 |
| 16 | 4.000 |
| 17 | 4.123 |
| 18 | 4.243 |
| 19 | 4.359 |
| 20 | 4.472 |
When you want to find the square root of a number that is not a perfect square (like 50), do the following:
There are several ways to find square roots:
To find the square root of a perfect square number by factorization:
Example 1: Find √144
144 = 2 × 2 × 2 × 2 × 3 × 3
Group into pairs: (2×2), (2×2), (3×3)
Take one number from each pair: 2 × 2 × 3 = 12
So √144 = 12
Example 2: Find √400
400 = 2 × 2 × 2 × 2 × 5 × 5
= (2×2) (2×2) (5×5)
Take one number from each pair: 2 × 2 × 5 = 20
√400 = 20
This method is used for larger numbers or non-perfect squares. It is a bit like ordinary long division but in a special pattern.
Steps (simplified):
Learning about square and square roots helps us to:
Example: If the area of a square field is 625 m², then the length of one side is √625 = 25 m.
This kind of reasoning is used in construction, land measurement, and science.
The study of squares and square roots is one of the most important topics in junior secondary mathematics. It builds the foundation for advanced topics in senior classes.
It also helps us solve everyday problems involving areas, distances, and estimations. By learning to use square and square root tables, students become faster and more confident in solving mathematical problems, especially when calculators are not allowed.
Understanding this topic prepares students for future mathematical studies and real-life applications.
A square garden has an area of 784 m². Find the length of one side of the garden.
Solution:
Area of a square = side × side = side²
So, side² = 784
Find the square root of 784:
√784 = 28
Therefore, each side of the garden is 28 metres long.
Find the area of a square tile whose side is 15 cm.
Solution:
Area = side × side = 15 × 15 = 225 cm²
The area of the tile is 225 cm².
The area of a square plate is 961 cm². Find the perimeter of the plate.
Solution:
side² = 961
√961 = 31
Perimeter of a square = 4 × side = 4 × 31 = 124 cm
Therefore, the perimeter = 124 cm.
A farmer wants to fence a square piece of land of 900 m². Find the total length of fencing wire he needs.
Solution:
side² = 900
√900 = 30
Perimeter = 4 × side = 4 × 30 = 120 m
The farmer needs 120 metres of wire.
Find the square root of 529 by factorization method.
Solution:
529 = 23 × 23 = (23 × 23)
Group factors: √(23 × 23) = 23
Therefore, √529 = 23.
(Or using factorization: 529 ÷ 23 = 23)
Find the square root of 784 by long division method.
Solution (Step-by-step):
Group digits in pairs from right to left → (7)(84)
Find the largest number whose square is ≤ 7 → that is 2 (2² = 4).
Subtract 4 from 7 → remainder 3. Bring down 84 → 384.
Double 2 → 4, now find a digit x such that 4x × x ≤ 384.
46 × 6 = 276
47 × 7 = 329
48 × 8 = 384
Therefore, √784 = 28.
The square root of 784 is 28.
Estimate √300 to one decimal place.
Solution:
The two nearest perfect squares to 300 are 289 and 324.
√289 = 17, √324 = 18
√300 lies between 17 and 18.
Now, 300 is closer to 289 than 324, so √300 ≈ 17.3
Therefore, √300 ≈ 17.3 (1 d.p.)
If the side of a square is increased from 12 cm to 15 cm, by how much does the area increase?
Solution:
Original area = 12 × 12 = 144 cm²
New area = 15 × 15 = 225 cm²
Increase = 225 − 144 = 81 cm²
Therefore, the area increases by 81 cm².
The square of a number is 1225. Find the number.
Solution:
Let the number be n.
Then, n² = 1225
Take square root of both sides:
n = √1225 = 35
The number is 35.
A square pool has a side of 18 m. Find:
(a) Its area, and
(b) Its diagonal length (correct to 1 decimal place).
Solution:
(a) Area = side² = 18² = 324 m²
(b) Diagonal = side × √2 = 18 × 1.414 = 25.5 m
Therefore, area = 324 m² and diagonal = 25.5 m.
