In this post we will try to understand the concept of mensuration and the ways to calculate the perimeter of different figures. Mensuration is very important and practical concept of mathematics and its questions are frequently asked in the examination, so make sure to invest proper time for preparation of this chapter.
Mensuration and Perimeter
What is mensuration?
Mensuration means “Measurement” The study of measurement of geometric figures and their parameters like length, Area, Volume etc. is called mensuration
What is Perimeter?
Perimeter of a shape is defined as the total distance around the shape. Basically, it is the length of any shape if it is expanded in a linear form
So, we can say that perimeter is the length of the wire needed to form the closed figure.
The idea of perimeter is widely used in our daily life
(01) A farmer owing a piece of land need to know the perimeter of the field in order to fence the whole field
(02) A person preparing a track to conduct sports should know the perimeter of the track required
Perimeter of any random shape = sum of length of all sides
Let us understand the concept with the help of examples
For this figure,
Perimeter = sum of all sides
= AB + BC + CD + DA
= 10 m + 8 m + 10 m + 8 m
= 36 m
Now let us look at another example
For this figure,
Perimeter = sum of all sides
= AB + BC + CD + DE + EF + FG + GH + HI + IJ + JK + KL + LA
= 2 m + 6 m + 6 m + 2 m + 6 m + 6 m + 2 m + 6 m + 6 m + 2 m + 6 m + 6 m
= 56 m
One more example
For this figure,
Perimeter = sum of all sides
= AB + BC + CD + DE + EF + FG + GA
= 4 m + 4 m + 3 m + 6 m + 5 m + 3 m + 3 m
= 28 m
Perimeter of some regular shapes
In this section we will try to understand the perimeter of common shapers which are regularly asked in the exams. I urge to understand and remember the formula for each of them as they are extremely important
Square
We can find the perimeter of square by adding all its sides.
In square all the sides are equal, so we can calculate perimeter of square by using following formula
Perimeter of square = 4 x length of each side = 4 x s
Solved Example
Q) Find the perimeter of a square whose side is 6 m.
Sol. Given, length of each side (s) = 6 m
Since, perimeter of square = 4 x length of each side (s)
= 4 x 6 m
= 24 m
Hence, perimeter of the square whose side is 6 m is 24 m. Ans.
Q) Find the side of the square whose perimeter is 20 m.
Sol. Given, Perimeter = 20 m
But we know that perimeter of a square = 4 x length of each side (s)
Therefore, 20 m = 4 x length of each side (s)
20/4 m = length of each side (s)
5 m = length of each side (s)
Hence, the length of side of square whose perimeter is 20 m is 5 m. Ans.
Rectangle
In rectangle, the adjacent side length is different, so the perimeter of rectangle can be calculated by adding all length of the sides.
Perimeter of Rectangle = 2 x (length of rectangle + breadth of rectangle)
= 2 x (L + B)
Solved Examples
Q) Find the perimeter of a rectangle whose length and breadth are 150 cm and 1 m respectively.
Sol. Given, Length = 150 cm
& Breadth = 1 m = 100 cm
Since, Perimeter of the rectangle = 2 × (length + breadth)
= 2 × (150 cm + 100 cm)
= 2 × (250 cm)
= 500 cm
= 5 m
Hence, perimeter of rectangle is 5 m. Ans.
Q) The perimeter of a rectangular park is 380m. Find the length the park if breadth is 80 m.
Sol. Given, Perimeter of rectangular park = 380 m
& breadth = 80 m
To find – length of rectangular park
Since, perimeter of a rectangle = 2 x (length + breadth)
Putting values,
380 m = 2 x (length + 80 m)
380/2 m = length + 80 m
190 m = length + 80 m
190 m – 80 m = length
110 m = length
Hence, length of rectangular park is 110 m. Ans.
Equilateral Triangle
In equilateral triangle, all the three sides are equal. So the perimeter of equilateral triangle is simple addition of all the three sides.
Perimeter of triangle = 3 x length of each side = 3 x s
Find the perimeter an equilateral triangle of side 9 cm.
Sol. Given, length of each side of an equilateral triangle (s) = 9 cm
Since, perimeter of an equilateral triangle = 3 x length of each side (s)
= 3 x 9 = 27 cm
Hence, perimeter of the equilateral triangle is 27 cm.
Rhombus
Perimeter of Rhombus = 4 x length of each side = 4 x s
Solved Examples
Q) Find the perimeter of rhombus of side 10 cm.
Sol. Given, side = 10 cm
Since, perimeter of rhombus = 4 x length of each side (s)
= 4 x 10
= 40 cm
Hence, perimeter of rhombus is 40 cm. Ans.
Parallelogram
In parallelogram, length of adjacent sides are different. The perimeter of parallelogram is again addition of all the sides.
Perimeter = 2 x (length of rectangle + breadth of rectangle)
= 2 x (L + B)
Solved Example
Q) Find the perimeter of a parallelogram whose length and breadth are 15 cm and 10 cm respectively.
Sol. Given, Length = 15 cm
& Breadth = 10 cm
Since, Perimeter of the parallelogram = 2 × (length + breadth)
= 2 × (15 cm + 10 cm)
= 2 × (25 cm)
= 50 cm
= 50 cm
Hence, perimeter of parallelogram is 50 cm.
Circle
In order to calculate the perimeter of circle you have to find the length of the circumference of circle.
Below is the formula for perimeter of circle. This formula is quite different from the one we discussed above, so make sure to remember it as it would help us to solve variety of problems.
Perimeter or circumference of circle = 2πr units = πd
Where, π = 22/7
Diameter = 2 x Radius
d = 2r
Here “r” represents the radius of a circle.
Solved Example
Q) Find the circumference/perimeter of a circle having radius = 7 cm.
Sol. Given, radius = 7 cm
Perimeter or circumference of circle, P = 2πr
P = 2 x 22/7 x 7 = 44 cm
Hence, perimeter of circle is 44 cm. Ans.
Q) Find the circumference/perimeter of a circle having diameter = 14 cm.
Sol. Given, diameter = 14 cm.
Therefore, radius = diameter = 14 = 7 cm
Perimeter or circumference of circle, P = 2πr
P = 2 x 22/7 x 7 = 44 cm
Hence, perimeter of circle is 44 cm. Ans.
Semi Circle
Perimeter or circumference of semi-circle = πr + 2r units = πr + d units
Where, π = 22/7
Q) Find the circumference/perimeter of a semi-circle having radius = 7 cm.
Sol. Given, radius = 7 cm
Perimeter or circumference of semi-circle, P = πr + 2r
P = 22/7 x 7 + 2 x 7
P = 22 + 14
P = 36 cm
Hence, perimeter of semi-circle is 36 cm. Ans
Cost Based Questions
Q) Find the cost of fencing a rectangular park of length 150 m and breadth 100 m at the rate of Rs.15 per meter.
Sol:
Given – Length of the rectangular park = 150 m
& Breadth of the rectangular park = 175 m
For calculating the cost of fencing we require perimeter.
Perimeter of the rectangle = 2 × (length + breadth)
= 2 × (150 m + 100 m)
= 2 × (250 m)
= 500 m
It is given that Cost of fencing of 1 m of park = Rs. 15
Therefore, the total cost of fencing the park = Rs. 15 × 500
= Rs. 7500
Hence, Cost of fencing the park is Rs. 7500
Q) Find the cost of fencing a square park of side 250 m at the rate of Rs.20 per meter.
Sol.
Given – length of each side of the square park (s) = 250 m
For calculating the cost of fencing we require perimeter.
Perimeter of the square = 4 x length of each side (s)
= 4 x 250 m
= 1000 m
It is given that Cost of fencing of 1 m of park = Rs. 20
Therefore, the total cost of fencing the park = Rs. 20 × 1000
= Rs. 20,000
Hence, Cost of fencing the park is Rs. 20,000 Ans.
Q) Find the cost of fencing a rectangular park of length 150 m and breadth 100 m at the rate of Rs.15 per meter.
Sol:
Given – Length of the rectangular park = 150 m
& Breadth of the rectangular park = 175 m
For calculating the cost of fencing we require perimeter.
Perimeter of the rectangle = 2 × (length + breadth)
= 2 × (150 m + 100 m)
= 2 × (250 m)
= 500 m
It is given that Cost of fencing of 1 m of park = Rs. 15
Therefore, the total cost of fencing the park = Rs. 15 × 500
= Rs. 7500
Hence, Cost of fencing the park is Rs. 7500 Ans.