Area of a Square

Area of a Square

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

Definition

A square is a parallelogram whose sides are congruent and whose interior angles are congruent.

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

A = a2

A: Area of a square
a: Side

Example 1

Find the area of the given square. Side: 4.

a = 4

So A = 42.

42 = 16

So A = 16.

Example 2

Find the area of the given square.

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.

32 = 9
(√2)2 = 2

So A = 9⋅2.

9⋅2 = 18

So A = 18.

Example 2: Another Solution

Find the area of the given square.

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.