Square Numbers and Square Roots

Learn what square numbers are, how to find them, and how to calculate square roots of perfect squares.

What Are Square Numbers?

A square number is the result of multiplying a whole number by itself.

$n^2 = n \times n$

For example: $4^2 = 4 \times 4 = 16$ , so $16$ is a square number.

The first twelve square numbers are:

$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$

They are called "square" numbers because you can arrange that many dots into a perfect square grid. For example, $9 = 3 \times 3$ dots arranged in a $3 \times 3$ square.

Note

Square numbers appear in real life when you calculate areas. A square room that is 5 m on each side has an area of $5^2 = 25 \text{ m}^2$ .

Square Roots

The square root of a number is the value that, when multiplied by itself, gives that number.

$\sqrt{n^2} = n$

For example: $\sqrt{49} = 7$ because $7 \times 7 = 49$ .

Key square roots to know:

Square	Square root
$\sqrt{1} = 1$	$\sqrt{36} = 6$
$\sqrt{4} = 2$	$\sqrt{49} = 7$
$\sqrt{9} = 3$	$\sqrt{64} = 8$
$\sqrt{16} = 4$	$\sqrt{81} = 9$
$\sqrt{25} = 5$	$\sqrt{100} = 10$

Example: What is $\sqrt{144}$ ?

Think: what number times itself equals $144$ ? $12 \times 12 = 144$ , so $\sqrt{144} = \mathbf{12}$ .

Tip

Learn the first 12 square numbers by heart. If you know $7^2 = 49$ , you automatically know $\sqrt{49} = 7$ . They are the same fact written backwards!

Working With Square Numbers

Example: Is $169$ a square number? Find its square root.

Try: $13 \times 13 = 169$ . Yes! $\sqrt{169} = \mathbf{13}$ .

Example: Calculate $3^2 + 4^2$ .

$3^2 = 9$ , $\quad 4^2 = 16$ , $\quad 9 + 16 = \mathbf{25}$

Notice: $25 = 5^2$ . The numbers $3, 4, 5$ form a famous right triangle!

Warning

$\sqrt{n}$ only gives a whole number when $n$ is a perfect square. $\sqrt{10}$ is not a whole number — it is approximately $3.16$ .

Real world: tiling a square courtyard

A square courtyard must be tiled with square tiles of side 1 m. If the courtyard is 9 m × 9 m, the total number of tiles needed is $9^2 = 81$ tiles. Knowing that $\sqrt{81} = 9$ also works backwards: if you have 81 tiles you can tile a 9 m × 9 m area exactly.

Real world: Pythagorean triples in construction

A builder uses the 3-4-5 right triangle ( $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ) to check corners are square. By measuring 3 m along one wall and 4 m along the adjacent wall, if the diagonal is exactly 5 m, the corner is a perfect right angle.

Tip The first 12 square numbers are worth memorising: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Knowing them instantly helps with area calculations and simplifying surds.

Practice Problems

Calculate $7^2$ and then find $\sqrt{49}$.

Show answer

$7^2 = 7 \times 7 = 49$. Therefore $\sqrt{49} = 7$.

Which square number lies between 50 and 60?

Show answer

$7^2 = 49$ (too small), $8^2 = 64$ (too big). Wait — there is no perfect square between 50 and 60. The nearest are 49 and 64.