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    Perimeter

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6th class > Mensuration > Perimeter

Perimeter

Looking at the above figures, we can see that from a random corner point, with the help of a wire or a string, if we move along the line segments we can reach the starting point again. Upon doing so, we complete one entire single round along the said shape in each case. The distance covered is equal to the length of cycle around the considered figure.

The distance covered during that one single cycle, is known as the perimeter of the closed figure, aka the length of the wire/string needed to form the figures.

Try to draw over the sides of the given below figures, with the red marker. The more accuartely the figure is marked, the closer you will be to the actual value of the perimeter of the figure.

The idea of perimeter is widely used in our daily life. Some examples which require perimeter calculations include-

  • A farmer who wants to fence his field.

  • An engineer who plans to build a compound wall on all sides of a house.

  • A person preparing a track to conduct sports.

All these people need to use the concept of ‘perimeter’.

Perimeter is the distance covered along the boundary forming a closed figure when we go round the figure once.

Now that we know about perimeter, how can we find the perimeter of any closed figure made up entirely of line segments? Take the figure below.

Perimeter = AB + BC + CD + DE + EF + FA = m

Simply find the sum of the lengths of all the sides (which are the line segments making it up).

Let's try some more of such problems!

Meera went to a park 150 m long and 80 m wide. She took one complete round on its boundary. What is the distance covered by her?

In the given below figure: we see that for one round of around the boundary of the park the distance is: + + +

= m

Let's continue.

Now, find the perimeter of the below square?

Perimeter = + + +

= cm

Find the perimeter of the given figure.

Perimeter = 1 + + + + + + + + + + +

= cm

Perimeter of a Rectangle

Let's move on to more specific and common geometrical shapes that are present in our everyday situations.

For example: Take the above rectangle ABCD whose length and breadth are 15 cm and 9 cm respectively. What will be its perimeter? Let's find out.

We know that in the case of a rectangle, the opposite sides are parallel and equal. Taking that property into account, we can say:

Perimeter of the rectangle = Sum of the lengths of its sides

= "ab + bc + cd + da" (Add up the labelled sides)

Taking the properties of a rectangle into consideration, we can take the common factor out of the brackets-

= 2 × ( ab + bc ) ( take AB-length and BC-breadth)

where ab = and bc =

= cm.

Note: that Perimeter of a rectangle = length + breadth + length + breadth and therefore, we get a general formula for the perimeter calculation of a rectangle i.e.

Perimeter of a rectangle = 2 × (length + breadth)

Let's practice some more problems.

Let's Practice the Logic

  • Shabana wants to put a lace border all around a rectangular table cover, 3 m long and 2 m wide. Find the length of the lace required by Shabana.

Understand the Problem:

Shabana wants to put a lace border around a rectangular table cover.

The dimensions of the table cover are given: length = meters, width = meters.

Identify What We Need to Find:

We need to find the total length of the lace required.

Since the lace goes around the table cover, we are essentially calculating the perimeter of the rectangle.

Formula for the Perimeter of a Rectangle:

The perimeter of a rectangle is calculated by adding all its sides together.

For a rectangle, Perimeter(P) = 2 x ( length + breadth )

Plug in the Values and Calculate:

Perimeter = 2 × (length+ breadth) = × ( + ) meters.

So, the length of the lace required by Shabana is meters.

An athlete takes 10 rounds of a rectangular park, 50 m long and 25 m wide. Find the total distance covered by him.

Length of the rectangular park = m

Breadth of the rectangular park = m

Total distance covered by the athlete in one round will be the of the park.

Perimeter of the rectangular park

= 2 × (length + breadth)= 2 × ( m + m)

= 2 × m

= m

So, the distance covered by the athlete in one round is 150 m.Therefore, distance covered in 10 rounds = × m

= m

The total distance covered by the athlete is 1500 m.

Find the perimeter of a rectangle whose length and breadth are 150 cm and 1 m respectively

Length = cm

Breadth = cm

Perimeter of the rectangle

= 2 × (length + breadth)

= 2 × ( cm + cm)

= 2 × cm

= cm = m

A farmer has a rectangular field of length and breadth 240 m and 180 m respectively. He wants to fence it with 3 rounds of rope as shown in figure 10.4. What is the total length of rope he must use?

The farmer has to cover three times the of that field.

Therefore, total length of rope required is its perimeter.

Perimeter of the field = 2 × (length + breadth)

= 2 × ( m + m)

= 2 × m

= m

Total length of rope required = × m = m

Find the cost of fencing a rectangular park of length 250 m and breadth 175 m at the rate of Rs. 12 per metre.

Length of the rectangular park = m

Breadth of the rectangular park = m

To calculate the cost of fencing we require perimeter.

Perimeter of the rectangle = 2 × (length + breadth)

= 2 × ( m + m)

= 2 × m

= m

Cost of fencing 1m of park = Rs. 12

Therefore, the total cost of fencing the park = Rs. 12 × 850 = Rs.

Perimeter of regular shapes

A regular shape is one which has

Take an example- We want to tape the boundary/border of a square picture of side 1m. What will be the length of the coloured tape we require?

Since this concerns the boundary, we need to find the of the picture frame.

Thus, the length of the tape required:

= Perimeter of square = m

Now, take an equilateral triangle with each side being 4 cm. The perimeter of this equilateral triangle = cm

So, the perimeter for an equilateral triangle becomes equal to times of the triangle. Now, what is similar between a square and an equilateral triangle? They are figures having all the sides of equal length and all the angles of equal measure. Such figures are known as regular closed figures. Thus, a square and an equilateral triangle are regular closed figures.

Perimeter of a square = 4 × of one side

Perimeter of an equilateral triangle = 3 ×

Similarly, perimeter of a regular pentagon = 5 × of one side and the perimeter of a regular hexagon will be times the and of an octagon will be times the .

Let's solve some problems

  1. Find the distance travelled by Shaina if she takes three rounds of a square park of side 70 m.

Understanding the Problem:

Shaina is walking around a square park. We know for a square all 4 sides are equal.

Side = m

Total number of rounds taken =

We need to calculate the total distance she travels.

Calculating the Perimeter of the Square:

Formula: The perimeter of a square

P = × side.

Calculation: Given the side of the square park is 70 meters, the perimeter

P = 4 × 70 meters = m

Shaina takes three rounds of the park. So, the total distance travelled = 3 × Perimeter of the park = 3 x 280 = m

Shaina travels a total of meters by taking three rounds of a square park with each side measuring 70 meters.
  1. Pinky runs around a square field of side 75 m, Bob runs around a rectangular field with length 160 m and breadth 105 m. Who covers more distance and by how much ?

Answer

Step 1: Define the Shapes and Measurements

Pinky is running around a square field.

Each side of Pinky's square field = m.

Bob is running around a rectangular field.

The length of Bob's field = 160m, and the breadth = m.

Step 2: Calculate the Perimeter of Pinky's Square Field

Perimeter of a Square = 4 × side

For Pinky: Perimeter = 4 × 75 m = m

Step 3: Calculate the Perimeter of Bob's Rectangular Field

Perimeter of a Rectangle = 2 × ( length + breadth )

For Bob: Perimeter = × ( + ) m = 2 x m = m

Step 4: Compare the Perimeters

Pinky's Distance: meters

Bob's Distance: meters

covers more distance.

Step 5: Calculate the Difference in Distances

Find the difference between the two perimeters to see by how much more one covers than the other.

Difference:

Bob's Distance-Pinky's Distance= 530 − 300 = m

Bob covers m more than Pinky.

  1. Find the perimeter of a regular pentagon with each side measuring 3 cm.

Solution:

Perimeter of regular polygon = Number of sides x Side Length

Perimeter of regular pentagon = x side length

= 5 x = cm

  1. The perimeter of a regular hexagon is 18 cm. How long is its one side?

Solution:

Perimeter of regular polygon = Number of sides x Side Length

Perimeter of regular hexagon = x side length

= 6 x side length

Length of side = cm