# Square: Area

How to find the area of a square (by using its properties): formula, 2 examples, and their solutions.

## Definition

## Formula

A = a^{2}

A: Area of a rhombus

a: Side

## Examplea = 4, A = ?

a = 4

So A = 4^{2}.

4^{2} = 16

So the area of the given square is 16.

## Example

A square is a rectangle.

So the segments formed by the diagonals

are all congruent: 3.

A square is also a rhombus.

So the diagonals are perpendicular.

Set the side of the given square a.

And see this right triangle.

The legs are both 3.

It's an isosceles right triangle.

So it's a 45-45-90 triangle.

So draw a 45-45-90 triangle

whose sides are 1, 1, and √2.

These two triangles are similar.

Then their sides are proportional.

So a/√2 = 3/1.

Similar Triangles

3/1 = 3

a/√2 = 3

Multiply √2 to both sides.

Then a = 3√2.

So the side of the given square is

a = 3√2.

a = 3√2

So A = (3√2)^{2}

(3√2)^{2}

= 3^{2}⋅(√2)^{2}

= 9⋅2

Power of a Product

9⋅2 = 18

So the area of the given square is 18.