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