Area of a Square

Area of a Square

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

Definition

A square is a parallelogram that have congruent sides and have congruent angles.

A square is a parallelogram
that have congruent sides and congruent angles.

It's a rectangle (angles) and a rhombus (sides).

So it has both properties.

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

A = 42

Example 2

Find the area of the given square. Segment from bisected diagonal: 3.

The square is a rhombus.

So the diagonals perpendicularly bisect each other.
So the brown segments are 3.

Diagonals of a rhombus

The bottom triangle is an isosceles right triangle.
So it's similar to a 45-45-90 triangle.

Draw a 45-45-90 triangle next to the given square.

The left triangle is × 3 bigger.

So the left triangle's hypotenuse is 3⋅√2.

a = 3√2

A = (3√2)2

Find the area of the given square. Segment from bisected diagonal: 3.

There's another way to solve this example.

The square is a rectangle.

So the blue segments formed by the diagonals
are all congruent: 3.

Diagonals of a rectangle

The square is also a rhombus.
And the diagonals are 6 and 6.

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