Rectangle is a closed two-dimensional figure composed of four sides and four vertices. All angles of the rectangle are 90 degrees. A rectangle with all sides equal is equivalent to a square. A rectangle is composed of two pairs of parallel sides, length, and width respectively.
Perimeter of Rectangle
The perimeter of a rectangle is the length of the outer boundary of a rectangle. It is also calculated by the summation of the total measure of both the lengths and breadths of the rectangle.
Perimeter of Rectangle Formula
Let us assume a rectangle of perimeter P, whose length and width are ‘l’ and ‘w’ respectively is 2(l + w).
Perimeter of a Rectangle Formula = 2 (Length + Width) units
How many different rectangles can be made with a perimeter of 24 cm?
Solution:
There may be multiple rectangles having a perimeter equivalent to 24 cm. For example, one rectangle of sides 5 cm and 7cm have perimeter 24, similarly, a rectangle of sides 4 cm and 8 cm have perimeter 24. So, there can be many rectangles of dimensions such as (5,7), (4,8), (3,9), (3.6,8.4), etc.
If a and b are the sides of the rectangle then the perimeter is 2a+2b. You are given a string 24 cm long (which will be the perimeter) from which to make the rectangles. So,
2a + 2b = 24
2(a + b) = 24
a + b = 12
If the sides must be integers, then the possibilities are:
a = 1, b = 11
a = 2, b = 10
a = 3, b = 9
a = 4, b = 8
a = 5, b = 7
a = 6, b = 6
If a=7, then b=5 which is the same as the fifth rectangle listed, i.e., a=5, b=7.
So there are only 6 possible rectangles (the square a=6 & b=6, being a special case of a rectangle).
Complete step by step solution:
We know,
The length of the string forming the rectangle is 24cm. Therefore, it can be concluded that the perimeter of the rectangle is equivalent to 24 cm.
The perimeter of a rectangle is given by 2(a+b)=P, which implies that the perimeter is equal to double of the sum of the sides of the rectangle.
Substituting, we get,
∴ 2(a+b) = 24
Therefore,
(a+b) = 12
Starting with a = 6, b = 6.
On increasing the value by 1, we conclude that, following are the other sides of the rectangle
a = 7, b = 5
a = 8, b = 4
a = 9, b = 3
a = 10, b = 2
a = 11, b = 1
Thus all we can say 6 rectangles are possible and the following are those rectangles:
- 6, 6, 6, 6
- 5, 7, 5, 7
- 4, 8, 4, 8
- 3, 9, 3, 9
- 2, 10, 2, 10
- 1, 11, 1, 11
Sample Questions
Question 1. Assume that the sides of the rectangle are in the ratio of 6:4 and its perimeter is 100 cm then find its dimensions?
Solution:
Here we have,
Perimeter of rectangle = 100 cm
Ratio of length and breadth is 6 : 4
We have to find length and breadth of the rectangle
Assume the common ratio be x
Thus, the sides will be 6x and 4x
As we know that,
Perimeter of the rectangle = 2(length + breadth)
Perimeter of the rectangle = 2(6x + 4x)
Perimeter of the rectangle = 2 × 10x
Perimeter of the rectangle = 20x
100 = 20x
x = 100/20
x = 5
Therefore,
Length = 6x = 6 × 5 = 30 cm
Breadth = 4x = 4 × 5 = 20 cm
Therefore,
Length is 30 cm and breadth is 20 cm.
Question 2. If the sides of the rectangle are in the ratio of 4:2 and its perimeter is 600 cm then find its dimensions?
Solution:
Here we have,
Perimeter of rectangle = 600 m
Ratio of length and breadth is 4 : 2
We have to find length and breadth of the rectangle
Assume the common ratio be x
Thus, the sides will be 4x and 2x
As we know that,
Perimeter of the rectangle = 2(length + breadth)
Perimeter of the rectangle = 2(4x + 2x)
Perimeter of the rectangle = 2 × 6x
Perimeter of the rectangle = 12x
600 = 12x
x = 600/12
x = 50
Therefore,
Length = 4x = 4 × 50 = 200 m
Breadth = 2x = 2 × 50 = 100 m
Therefore,
Length is 200 m and breadth is 100 m.
Question 3. Find how many tiles of dimensions length 26 cm and breadth 14 cm will be required to cover a rectangular godown floor of dimensions length 1040 cm and breadth 280 cm?
Solution:
To find the number of tiles
First we need to find the area of tile and godown
Area of rectangle = Length × Breadth
Area of rectangle tile = 26 × 14
Area of rectangle tile = 364 cm2
Area of rectangle godown = 1040 × 280
Area of rectangle godown = 291200 cm2
Now,
Finding the number of tiles required
Number of tiles = Area of godown/ Area of a tile
Number of tiles = 291200/364
Number of tiles = 800 tiles
Therefore,
We will be needing 800 tiles to cover the godown floor.
Question 4. Find the cost of fencing a rectangular playground with an area of 3200 m2 at the rate of ₹20 per meter? If the length is double the breadth.
Solution:
Here we have to find the cost of fencing the rectangular playground
As given in the question, length is double the breadth
Breadth = b
Length = 2 × b = 2b
First we need to find the length and breadth of the playground
Area of rectangle = Length × Breadth
Area of rectangular playground = 2b × b
3200 = 2b2
b2 = 3200/2
b2 = 1600
b = √1600
b = 40
Breadth = 40 m
Length = 2 × breadth = 2 × 40 = 80 m
Further,
Perimeter of the rectangle = 2(length + breadth)
Perimeter of the rectangle = 2(80 + 40)
Perimeter of the rectangle = 2 × 120
Perimeter of the rectangle = 240 m
Now,
Finding the cost of fencing
Cost of fencing the playground = ₹20 × perimeter of the playground
Cost of fencing the playground = ₹20 × 240
Cost of fencing the playground = ₹480
Therefore,
Cost of fencing the rectangular playground is ₹480.