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.

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.

## Area of a Square

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.

### Rectangle

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