Square: Area

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

Definition

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.

Formula

Formula

A = a2

A: Area of a rhombus
a: Side

Example 1

Example

Solution

a = 4

So A = 42.

42 = 16

So the area of the given square is 16.

Example 2

Example

Solution

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
= 32⋅(√2)2
= 9⋅2

Power of a Product

9⋅2 = 18

So the area of the given square is 18.