A square is a regular quadrilateral. A square is a four sided figure which has four equal sides and four equal angles. Each angle is $\frac{360^o}{4}$ = 90° and so every angle of a square is a right angle. Opposite sides of a square are parallel. The length of a side of the square is the distance between any two adjacent vertices or corners. A square with vertices or corners as A, B, C and D would be denoted  by $\square$ABCD.
Square Definition 
In this figure on the left A, B, C and D are the vertices or corners of $\square$ABCD. AB, BC, CD and AD are the four sides of $\square$ABCD.

Like for most quadrilaterals, the area of a quadrilateral is the length of one side times its perpendicular height. The length of a side of the square is the distance between any two adjacent vertices or corners. Since all the sides in a square are equal, the height of the square is same as the length of the side of a square. Once the length of a square is found, the area of the square is found by multiplying the side length by itself. The formula is given by
Area of a square = $s^2$, where s is the length of any side.

Solved Examples

Question 1: Let the side length of a square be 6 cms. Find the area of the square.
Solution:
 
The length of the side of a square, s = 6 cms.
Area of the square = $s^2$ = (6)$^2$   = 36 sq.cms.
 

Question 2: The area of square is 144$cm^2$. What is the side length of the square?
Solution:
 
The area of the square = $s^2$  = 144$cm^2$

The side length of the square, s = $\sqrt{area}$ = $\sqrt{144}$ = 12 cms.
 

Question 3: The area of a square increases by 56 feet$^2$ when the sides of a square are each increased by 2 feet. Find the length of each side S before the increase.
Solution:
 
Let the length of the side of the original square be s, then the area of the original square is given by $A_1$ = $s^2$.

Given that the side of the square is increased by 2 feet. That means s becomes s+2 for the changed new square. Thus the area of changed square is given by, $A_2$ = (s+2)$^2$.

Also given, area of the square increases by 56 feet. Thus we have. $A_2$ = $A_1$ + 56
                                  (s + 2)$^2$ = $A_1$ + 56
                                  (s + 2)$^2$  = $s^2$ + 56
                                  s$^2$ + 4s + 4 = $s^2$ + 56
                               s$^2$ + 4s - s$^2$ = 56 - 4
4s = 52
s = 13 feet.
The length of each side of the square before increase is 13 feet.
 

In Geometry, the Perimeter of any polygon is the total distance around the outside of a 2-dimensional shape. All the four sides of a square are congruent. So, the perimeter of a square is four times its side length. The formula is given by
Perimeter of a square = 4s, where s is the length of any side.
A square has a larger area than any other quadrilateral with the same perimeter.

Solved Examples

Question 1: Let the side length of a square be 6 cms. Find the perimeter of the square.
Solution:
 
The length of the side of a square, s = 6 cms.

Perimeter of the square = 4s= 4(6) = 24 cms.
 

Question 2: The perimeter of square is 144 cms. What is the side length of the square?
Solution:
 
The perimeter of the square = 4s = 144 cms.

The side length of the square, s = $\frac{perimeter}{4}$ = $\frac{144}{4}$ = 36 cms.
 

Question 3: The area of square is 144 cm$^2$. What is the perimeter of the square?
Solution:
 
The area of the square = $s^2$ = 144 cm$^2$
The side length of the square, s = $\sqrt{area}$ = $\sqrt{144}$ = 12 cms.
The perimeter of the square = 4s = 4(12) = 48 cms.
 

In a polygon any line joining two non-consecutive vertices is called a diagonal. The distance between any pair of opposite vertices gives us the length of a diagonal. In the figure below, AC and BD are the diagonals of the square ABCD.

Diagonal of a Square

In a square, the length of each diagonal is given by s$\sqrt{2}$, where s is the length of any side.

Properties of diagonals of a square:
1. The diagonals of a square are perpendicular bisector of each other.
2. The diagonals of a square bisect its angles at the vertex.
3. The lengths of the two diagonals of a square are equal.
4. Each diagonal divides the square into two congruent isosceles right triangles. And so they have the same area, which is half the area of the square.
Example: The side length of square is 12 cms. What is the length of each diagonal of the square?
Solution: The side length of the square = s = 12cms.
The length of a diagonal of the square, d = s$\sqrt{2}$ =(12)$\sqrt{2}$ = 12 $\times$ 1.414 = 16.97 cms.

Area of a square using diagonal:
If the length of a diagonal is known, the area is given by, area = $\frac{d^2}{2}$

area = $\frac{d^2}{2}$, where d is the length of either diagonal.

Example: Find the area and perimeter of a square with diagonal of 20 meters.
Solution: The length of the diagonal, d = 20 meters.

The area of the square = $\frac{d^2}{2}$ = $\frac{20^2}{2}$ = 200 meters$^2$.

The side length of the square, s = $\sqrt{area}$ = $\sqrt{200}$ = 14.14 meters.
The perimeter of the square = 4s = 4(14.14) = 56.56 meters.
The area of the given square is 200 meters$^2$ and the perimeter is 56.56 meters.