Can a square and a rectangle have the same area and perimeter?
Mensuration is the branch of mathematics that deals with the calculation of parameters like weight, volume, area, perimeter, etc. of geometrical shapes. Mensuration is concerned with 2D and 3D shapes.
In 2D shapes, objects have two dimensions and are placed on a plane surface. The two-dimensional will be only the length and breadth of the object. It doesn’t deal with the thickness of the object.
3D shapes of objects have three dimensions and are present in the real world. The three dimensions are height, width, and depth.
Formulas of 2D shapes
Some standard formulas for the calculation of parameters of some two-dimensional shapes are given below:
- Rectangle
Perimeter = 2(length + breadth)
Area = length × breadth
- Square
Area = (side)2
Perimeter = 4(side)
- Circle
Diameter = 2 × radius
Area = πr2
- Triangle
Area = 1/2 base × height
Formulas of 3D shapes
Some standard formulas for the calculation of parameters of some three-dimensional shapes are given below:
- Cube
Volume = (side)3
Lateral surface area = 4(side)2
Total surface area = 6(side)2
- Cuboid
Volume = length × width × height
Lateral surface area = 2h(l+b)
Total surface area = 2(lb+lh+bh)
- Sphere
volume = 4/3πr3
Surface area = 4πr2
- Cone
Volume = 1/3πr2h
Total surface area = πr(l+radius)
Can a square and a rectangle have the same area and perimeter?
Evaluating area of square and rectangle.
According to the question we have to find out whether the area of two geometrical shapes square and rectangle are equal or not.
Let’s assume a square with its side length be l. And, a rectangle with length (l) and breadth (b).
Now,
Area of square = Area of rectangle
Using the standard formulas,
=>(side)2 = length × breadth
=>l2 = l × b
=>l = b
If b = 4, then, l = 4
Calculating the area of square and rectangle
Area of square = (side)2
= (l)2 = l × l = 4 × 4 = 16
Area of rectangle = l × b
= 4 × 4 =16
Hence, the area of the square and rectangle can be the same.
Following the same method let’s evaluate the perimeter of a square and a rectangle.
Evaluating the perimeter of a square and rectangle
Let’s assume a square with its side length be l. And, a rectangle with length (l) and breadth (b).
Now,
Perimeter of square= perimeter of rectangle
=>4(side) = 2(length+breadth)
=>4l = 2(l+b)
=>4l = 2l+2b
=>4l-2l = 2b
=>2l = 2b
=>l = b
If l = 4 then, b = 4
Calculating perimeter of square = 4(side)
= 4l = 4 × 4 = 16
Calculating perimeter of rectangle = 2(l+b)
= 2(4+4) = 2 × 8 = 16
Hence, a square and rectangle can have the same area and perimeter in certain conditions.
Sample Questions
Question 1. Find the area of a triangle having a base of 9cm and a height of 5 cm.
Solution:
Given,
Base of triangle(b) = 9cm
Height of triangle(h) = 5cm
Now,
Area = 1/2 b × h
=>1/2 × (9 × 5)
=>45/2cm2
Question 2. The area of a rectangle is 48cm2. If it has a length of 8cm find its breadth.
Given,
Length (l) = 8cm
Area(A) = 48cm2
Now,
Area = l × b
=>48 = 8 × b
=>b = 48/8
=>b = 6cm
Question 3. Find the area of a square having one side 15cm.
Solution:
Given,
Side(l) = 15cm
Now,
Area = (side)2
= l2 = 15 × 15 = 225cm2
Question 4. If a rectangle has a length of 10cm and a breadth of 5 cm. What will be its perimeter?
Solution:
Given,
Length(l) =10cm
Breadth(b) = 5cm
Now,
Perimeter = 2(l+b)
=>P = 2(10+5)
=>P = 2 × 15
=>P = 30cm
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