Perimeter and Area

Perimeter and Area

Recognition of Region of a Closed Figure

Region of a closed figure comprises of the surface enclosed by the boundary and the boundary of the figure itself.

The regions of the different closed figures are given below:

Differentiation Between Perimeter and Area of a Region

Perimeter

Perimeter is the distance around a closed figure. In other words the length of the boundary of the closed figure is known as the perimeter of the figure. Since the perimeter is a distance, it is measured in cm, m and km. Area is the quantity that expresses the extent of a two dimensional figure. Area is measured in square units i.e.\(cm^2\), \(m^2\) and \(km^2\) .

Perimeter and Area

Perimeter

(i)Perimeter of a square

We know that a square has four sides of equal length. To find the perimeter of the square, we add the lengths of four sides of a square

i.e; Perimeter of a square = side + side + side + side = 4 × side

∴ Formula for perimeter of a square = 4 × side units

Example:

Find the perimeter of a square whose length of a side is 5 cm.

Solution:

Length of a side = 5 cm

Perimeter of the square= 4 × side

=4 × 5

=20cm

(ii) Perimeter of a Rectangle

We know that a rectangle has 2 equal lengths and 2 equal breadths. To find the perimeter of a rectangle, we add the measures of four sides i.e.,

Perimeter of the rectangle= Length + Breadth + Length + Breadth

=Length + Breadth + Length + Breadth

=2 (Length) + 2(Breadth)

=2 (Length + Breadth)

∴ Formula for perimeter of a rectangle = 2 (Length + Breadth) units

Example:

Find the perimeter of a rectangle whose length is 5cm and breadth is 4cm.

Solution:

Length of the rectangle=5 cm

 Breadth of the rectangle=4 cm

 Perimeter of the rectangle =2 (Lenth + Breadth)

= 5(5 + 4) =2(9) =18 cm

Perimeter and Area

Area of a Square and Rectangle

 (i) Area of a Square

The area is the product of length and breadth.

In a square length of each side is equal i.e.

 length = breadth = side

Area of a square = side × side

∴ Formula for the area of a square = side × side (units)\(^2\)

Example:

Find the area of a square whose length of a side is 3 cm.

Solution:

Length of side = 3 cm

 Area = side × side

= 3 × 3 = 9 cm\(^2\)

(ii) Area of a Rectangle

 The area of a rectangle can be calculated with the help of product of length and breadth.

 ∴ Formula for the area of a rectangle Length × Breadth (units)\(^2\)

Example: Find the area of a rectangle whose length is 12 cm and breadth is 8 cm.

Solution:

Length of the rectangle = 12 cm

Breadth of the rectangle = 8 cm

 Area of the rectangle = Length × Breadth

= 12 × 8 = 96 cm\(^2\)

Perimeter and Area

Application of formulas to find Perimeter and Area of a Square and a Rectangular Region

Example 1:

Find the perimeter and area of a square whose side is 12 cm.

Solution:

Length of the side = 12 cm

Perimeter of the square = 4 × side

= 4 × 12 =48cm

Area of the square = side × side = 12 × 12 = 144 cm\(^2\)

Example 2:

 Find the perimeter and area of a rectangle whose length is 12 cm and breadth is 8 cm.

Solution:

Length of the rectangle = 12 cm

Breadth of the rectangle = 8 cm

Perimeter of the rectangle = 2(Length + Breadth)

= 2(12 + 8) = 2(20)

= 40 cm

Area of the rectangle = Length × Breadth

= 12 x 8 = 96 cm\(^2\)

Perimeter and Area

Solution of Appropriate Problems of Perimeter and Area

Example 1:

The length of a square shaped room is 5 metre. Find the cost of flooring at the rate of Rs.900 per square metre

Solution:

Length of the room = 5 m

Area of the room = side × side

= 5 × 5 = 25 m\(^2\)

Cost of flooring 1 m = Rs. 900

Cost of flooring 25 m = 25 × 900

The cost of fencing 68 m = 68 × 10 = Rs. 22,500

Example 2:

The Length of the side of a square shaped field is 17 m. Find the cost of fencing it at the rate of Rs. 10 per meter.

Solution:

Perimeter of the field = 4 × side

= 4 × 17 = 68 m

The cost of fencing 1 m = Rs. 10

The cost of fencing 68 m = 68 × 10

= Rs. 680

Example 3:

The length of a rectangular field is 120 m and its breadth is 80 m. Find the cost of:

(a) Fencing it at the rate of Rs. 100 per metre and

(b) Ploughing it at the rate of Rs. 10 per square metre

Solution:

(a)Length of the field = 120 m

(b)Breadth of the field = 80 m

 Perimeter of the field = 2(Length + Breadth)

= 2(120 + 80) = 2(200) = 400 m

The cost of fencing 1 m = Rs. 100

The cost of fencing 400 m = 400 × 100

= Rs. 40,000

(b) Area of the field = length × breadth

= 120 × 80

= 9600 m. The cost of ploughing 1 m = Rs. 10

The cost of ploughing 9600 m = 9600 × 10 = Rs. 96,000

Example 4:

The perimeter of a square shaped field is 20m. Find the area of the field.

Solution:

Perimeter of the field = 20 m

4 × side = 20 m

side = \(20\over 4\) = 5 m

Area of the field = side × side

= 5 × 5 = 25 m\(^2\)

Example 5:

The perimeter of a rectangular orchard is 250 m.

 If the length is 75 m, find the breadth of the orchard.

Solution:

Perimeter of the orchard = 250 m

 i.e. 2(Length + Breadth) = 250 m

 ∴ Length + Breadth = 125 m

Length of the orchard = 75 m

∴ 75 + Breadth = 125 = 125

⇒ Breadth = 125 – 75 = 50 m