Area and Perimeter

When we talk about some plane figures we think about their boundaries and regions. To know the measurements of the boundaries and regions of the shapes we use ‘Mensuration’ methods to get the desired result.

The Mensuration methods which are used to find the boundaries and regions of the different shapes are as follows:

• Perimeter
• Area

Now we discuss how the above two methods are used in finding the boundaries and regions of the different shapes.

Perimeter

• You can make the shapes using wires or a string.
• If you start at one point, make any shape with the wire or a string and reach the same point and form the round shapes in each case.
• The distance covered is equal to the length of the wire used to form the shapes.
• This distance is known as the perimeter of the closed shape. It is the length of the wire or a string that need to form a shape.
• Perimeter idea is widely used in daily life like
  • A farmer who wants to fence a field.
  • Engineers who want to build a compound wall around the house.
  • The person preparing a track to conduct the sports.

• Perimeter is the distance covered along with the boundary forming a closed shape 

Perimeter of Rectangle

Let us consider a rectangle ABCD whose length and breadth are 5 and 10 cm respectively. What will be their perimeter?

The perimeter of the rectangle is equal to the sum of the length of its four sides.

Opposite sides are equal like, AB = CD and AD = BC. Length of the four sides is as follow:

• AB + BC + CD + DA
• AB + BC + AB + BC
• 2 × AB + 2 × BC
• 2 × ( AB + BC)
• 2 × ( 5 + 10)
• 2 × ( 15 cm)
• 30 cm

Hence from the above example, we can tell that, 

Perimeter of Rectangle = length + breadth + length + breadth.

I.e. Perimeter of Rectangle = 2 × (length + breadth).

Let’s see a few practical solutions for the perimeter of a rectangle.

Example 1-

An athlete takes 10 rounds of the rectangular park, 20 m long and 15 m wide. Find the distance covered by him?

Solution:

Let,

Length of the rectangular park = 20 m.

The breadth of the rectangular park = 15 m.

The total distance covered by an athlete in one round will be the perimeter of the park.

Now, the perimeter of the rectangular park

• 2 × (length + breadth)
• 2 × (20 + 15)
• 2 × (35 m)
• 70 m.

So distance covered by an athlete in one round is 70m

Then, the distance covered by an athlete in 10 rounds = 10 ×70 = 700 m.

The total distance covered by an athlete is 700 m. 

Example 2:

Find the perimeter of the rectangle whose length and breadth is 45 cm and 25 cm respectively.

Solution:

Let,

Length of the given rectangle = 45 cm.

Breadth of the given rectangle = 25 cm.

Perimeter of the Rectangle

• 2 × (length + breadth)
• 2 × (45 + 25)
• 2 × (70 cm)
• 140 cm.

Perimeter of a Square

Let ABCD be the four sides of the square then, 

Perimeter of the Square = 4 × length of the sides.

Example:

Find the distance travelled by alia is she takes 3 rounds of a square park of side 150 m.

Solution:

Let,

Length of the given square = 150 m.

Perimeter of the Square

• 4 × length of the sides
• 4 × 150 m
• 600 m.

The distance travelled by alia of a square park in round is 600 m.

Then, the distance travelled by alia of a square park in 3 rounds = 600 × 3 = 1600 m.

Perimeter of Equilateral Triangle

In equilateral triangle all sides and angles are equal. So perimeter of the equilateral triangle is equal to the multiplication of all 3 sides.

Perimeter of the Equilateral Triangle = 3 × length of sides.

Perimeter of the Pentagon

In the pentagon all 5 sides and angles are equal. So perimeter of the pentagon is equal to the multiplication of all 5 sides.

Perimeter of the pentagon = 5 × lengths of sides.

Area

• The amount of region enclosed by a closed shape is called its Area.
• In the square is enclosed by the shapes. Some of them are completely enclosed, a few are half enclosed, a few less than half and a few more than half enclosed.
• There is a tiny problem; the square do not always fit exactly into the area you measured.
• We can get over this problem by adopting the convention method.
  • The area of one full square unit is taken as 1 sq unit. If it is a centimetre square sheet then, the area of one full square is taken as 1 sq cm.
  • Ignore the portion of the area that is less than half of the square region.
  • If more than have is in the square region then count it as a full square.
  • If exactly half of the area of the square region is counted then take its area as ½ sq units.
  • Such a convention gives the fair estimation of the desired area.

Area of Rectangle

With help of a square paper can you tell what will be the area of the rectangle whose length is 3 cm and breadth is 5 cm?

• • Draw the rectangle on a graph having 1 cm × 1 cm squares.
  • Make sure the rectangle covers 15 squares completely.
  • The area of the rectangle is 15 sq cm which can be written as 3 × 5 sq cm i.e. (length × breadth).

• Area of a Rectangle  = (length × breadth)

Area of Square

• Let us consider a square side of 5 cm.
• If we place the square in the graph paper then it covers 25 squares i.e. area of the square is equal to 25 sq cm = 5 × 5.
• Area of the Square = side × side.

Example 1:

Find the area of a rectangle whose length and breadth are 10 and 5 cm respectively.

Solution:

Let’s consider,

Length of the given rectangle = 10 cm.

Breadth of the given rectangle = 5 cm.

Area of the rectangle

• Length × breadth

• 10 × 5
• 50 cm.

Example 2:

Find the area of the square plot whose side is 12 m.

Solution:

Let’s consider,

Length of the square plot = 12 m.

Area of the square

• Side × side

• 12 ×12
• 144 m.

Conclusion:

Here, we learned about what is the perimeter, when it is used and also how it used in daily life. We also learned about what is an area and how they are used in daily life. We have talked in detail about various shapes of figures and how to calculate the perimeter of each of them. Likewise, we have talked about the area of the figures and how to calculate them. By memorizing the formula it will become easy to solve the mathematical problems.