AREA OF SQUARES AND RECTANGLES

Square

A square is a four-sided closed figure where the lengths of all the four sides will be equal and each vertex angle will be right angle or 90^o as shown below. 

Formula for Area of a Square :

Let s be the length of each side of a square. 

Then, the formula for area of a square : 

Area  =  s²

Let d be the length of each diagonal of a square. 

Then, the formula for area of a square : 

Area  =  1/2 ⋅ d²

Rectangle

A rectangle is a four-sided polygon where the lengths of opposite sides will be equal and each vertex angle will be 90° or right angle as shown below

Formula for Area of a Rectangle :

To get the area of any rectangle, we have to multiply its length and width.  

Let l be the length and w be the width of a rectangle.

Then, the formula for area of the rectangle : 

Area  =  l ⋅ w

Examples

Example 1 :

Find the area of the square having side length 24 cm.

Solution:

When the length of a side is given, formula for area of a square :

=  s² 

Substitute 24 for s.

=  24²

=  576

So, area of the square is 576 square cm.

Example 2 :

If the length of each diagonal is 2√2 cm, then find its area.

Solution:

When the length of a diagonal is given, formula for area of a square :

=  1/2 ⋅ d²

Substitute 2√2 for d.

=  1/2 ⋅ (2√2)²

Simplify.

=  1/2 ⋅ (4 ⋅ 2)

=  1/2 ⋅ (8)

=  4

So, the area of the square is 4 square cm. 

Example 3 : 

Find the area of a rectangle, if its length is 15 cm and width is 20 cm.

Solution : 

Formula for area of a rectangle :

=  l ⋅ w

Substitute 15 for l and 20 for w.

=  15 ⋅ 20

=  300

So, the area of the rectangle is 300 square cm.

Example 4 : 

The length of a rectangle is 3 times its width. If its perimeter is 32 ft, then find the area of the rectangle.  

Solution : 

Let x be the width of the rectangle.

Then, its length is 3x. 

Perimeter of the rectangle is 32 ft

2(l + w)  =  32

Divide each side by 2. 

l + w  =  16

Substitute 3x for l and x for w.

3x + x  =  16

4x  =  16

Divide each side by 4. 

x  =  4

Therefore, width of the rectangle is 4 ft. 

And length of the rectangle is

=  3(4)

=  12 ft

Formula for area of a rectangle :

=  l ⋅ w

Substitute 12 for l and 4 for w.

=  12 ⋅ 4

=  48

So, the area of the rectangle is 48 square ft. 

Example 5 : 

The floor of a room is in rectangular shape and it has 13 m length and 9 m width. If the cost of the carpet is $12.40 per square meter, find the total cost of carpeting the floor of the room. 

Solution : 

To find the total cost of carpeting the floor of the room, we have to know its area. Because the floor of the room is in rectangle shape, we can use the formula for area of a rectangle to find the area of the floor.    

Formula for area of a rectangle :

=  l ⋅ w

Substitute 13 for l and 9 for w.

=  13 ⋅ 9

=  117

So, the area of the floor is 117 square meters.

The cost of carpet is $12.40 per square meter. 

Then, the total cost of carpet for 117 square meters : 

=  117 ⋅ 12.40

=  1450.80

So, the total cost of carpeting the floor of the room is $1450.80.

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