# Area of a Square

How to find the area of a square: definition, formula, examples, and their solutions.

## Definition

A square is a quadrilateral

whose side are congruent

and whose interior angles are congruent (= 90º).

It's a rectangle and a rhombus.

So it has all the properties

of a rectangle and a rhombus.

Properties of a rectangle

Properties of a rhombus

## Formula

*A* = *a*^{2}*A*: Area of a square*a*: Side

## Example 1

*a* = 4

So *A* = 4^{2}.

4^{2} = 16

So *A* = 16.

## Example 2

A square is a rectangle.

And for a rectangle,

the segments formed by the diagonals

are all congruent.

Properties of a rectangle - Diagonals

So the lower right segment is 3.

A square is also a rhombus.

And for a rhombus,

the diagonals perpendicularly bisect each other.

So the diagonals are perpendicular.

So draw a right angle like this. (brown)

Properties of a rhombus - Diagonals

See this right triangle.

Set the bottom side as [*a*].

Then its sides are 3, 3, and [*a*].

This right triangle is an isosceles right triangle.

So this triangle is a 45-45-90 triangle.

45-45-90 triangle

Draw a 45-45-90 triangle,

whose sides are 1, 1, and √2,

next to the given square.

Then these two triangles are similar.

Since these two triangles are similar,

their sides are proportional.

So *a*/√2 = 3/1.

Similar triangles

Multiply √2 on both sides.

Then *a* = 3√2.

*a* = 3√2

So *A* = (3√2)^{2}.

3^{2} = 9

(√2)^{2} = 2

So *A* = 9⋅2.

9⋅2 = 18

So *A* = 18.

## Example 2: Another Solution

Let's see another way to solve this example.

A square is a rectangle.

And for a rectangle,

the segments formed by the diagonals

are all congruent.

So the blue segments are all 3.

A square is also a rhombus.

The diagonals are, 3 + 3, 6.

So the area of the rhombus is*A* = (1/2)⋅6⋅6.

Area of a rhombus

(1/2)⋅6 = 3

3⋅6 = 18

So *A* = 18.

As you can see,

you can get the same answer.