What happens to the area of a square when the perimeter is doubled?

Square is a geometrical figure which is formed of four equal sides. It is a regular quadrilateral where all the angles are right angles. Basically, it can be considered as a special case of a rectangle, where all the sides are equal in length. For instance, a square courtyard.

Properties of square

• All four sides of a square are equal.
• The diagonals of a square are equal.
• All the four angles of a square are equal.
• The diagonals of a square bisect its angles.

Perimeter of a square

The perimeter of a square is referred to as the total length of the boundary enclosing the geometrical figure.

Perimeter of square = Sum of all sides of a square

Let us assume s to be the side of the square.

Since, we know, all the sides of the square are equal.

Perimeter of square = s + s + s + s

Perimeter of square = 4s ….. (I)

Area of a square 

The area of a square is the space enclosed within its boundaries. It is described in sq. units each. 

Area of square = Product of any two sides

Let us assume s to be the side of the square.

Since, we know, all the sides of the square are equal.

Area of square = s * s

Area of square = s² ….. (I)

What happens to the area of a square when the perimeter is doubled?

Solution: 

Now, 

Let us assume P to be the perimeter of the square. Now, as proved above,

P = 4s, where s is the side of the square….I

According to the question, 

Perimeter is doubled. Let us assume the new perimeter to be P’ and s’ to be the new side of the square. 

P’ = 2P

Substituting values of P from I, we have

P’ = 2 (4s)

P’ = 8s

Therefore, 

s = P’/8

Now, 

P’ = 4s’

Substituting the values, 

s = 4s’/8

s = s’/2

s’ = 2s 

Therefore, the side becomes twice the original side. 

Now,

Old Area, A = s²

New Area, A’ = s’²

Substituting values of s’ from II,we get, 

A’ = (2s)²

A’ = 4s²

Therefore, 

A’ = 4A

Thus, area becomes 4 times of the original area. 

Sample Questions

Question 1. If the perimeter of a square is 16 cm then find the area of the square and also find how many times the area of the square will increase if the perimeter is doubled?

Solution:

Here we have the perimeter of square 16 cm

As we know that

Perimeter of square = 4 × side (s)

16 = 4 × side

s = 16/4

s = 4 cm

Side of a square is 4 cm

Further,

Finding the area of square 

Area of square = Side × Side

Area of square = 4 × 4

Area of square = 16 cm²

Now,

When the perimeter of the square is doubled

2 × Perimeter of the square = 2 × 16

= 32 cm

Now,

Side of the doubled perimeter = s’

32 = 4 × s’

s’ = 32/4

s’ = 8 cm

Further new area

Area of doubled perimeter square = s’ × s’

Area of double perimeter square = 8 × 8

Area of double perimeter square = 64 cm²

To find how many times the area of square increased after doubling the perimeter

= 

= 64/16

= 4 times

Therefore,

Area of the square is increased by 4 times after doubling the perimeter of the square.

Question 2. Find how many times the area of the square is increased if its perimeter is doubled? If the side of the square is 10 m.

Solution:

Here we have the side of square 10 m

As we know that

Perimeter of a square = 4 × side (s)

Perimeter of a square  = 4 × 10

Perimeter of a square = 40 m

Perimeter of a square = 40 m

Perimeter of a square = 40 m

Further,

Finding the area of square

Area of square = Side × Side

Area of square = 10 × 10

Area of square = 100 m²

Now,

When the perimeter of the square is doubled

2 × Perimeter of the square = 2 × 40

= 80 m

Now,

Side of the doubled perimeter = s’

80 = 4 × s’

s’ = 80/4

s’ = 20 m

Further new area

Area of doubled perimeter square = s’ × s’

Area of double perimeter square = 20 × 20

Area of double perimeter square = 400 m²

To find how many times the area of square increased after doubling the perimeter

= 

= 400/100

= 4 times

Therefore,

Area of the square is increased by 4 times after doubling the perimeter of the square.

Question 3. Calculate how many times the area of the square is increased if the perimeter of the square is tripled? The perimeter of the square is 20 cm.

Solution:

Here we have the perimeter of square 20 cm

As we know that

Perimeter of square = 4 × side (s)

20 = 4 × side

s = 20/4

s = 5 cm

Side of a square is 5 cm

Further,

Finding the area of square

Area of square = Side × Side

Area of square = 5 × 5

Area of square = 25 cm²

Now,

When the perimeter of the square is tripled

3 × Perimeter of the square = 3 × 20

= 60 cm

Now,

Side of the doubled perimeter = s’

60 = 4 × s’

s’ = 60/4

s’ = 15 cm

Further new area

Area of tripled perimeter square = s’ × s’

Area of tripled perimeter square = 15 × 15

Area of tripled perimeter square = 225 cm²

To find how many time area of square increased after tripling the perimeter

= 

= 225/25

= 9 times

Therefore,

Area of the square is increased by 9 times when the perimeter of the square is tripled.

Question 4. The perimeter of the square is 40 m. If the perimeter of the square is increased by 4 times then find how many times the area of the square is increased?

Solution:

Here we have the perimeter of square 40 cm

As we know that

Perimeter of square = 4 × side (s)

40 = 4 × side

s = 40/4

s = 10 m

Side of a square is 10 m

Further,

Finding the area of square

Area of square = Side × Side

Area of square = 10 × 10

Area of square = 100 m²

Now,

When the perimeter of the square is 4 times,

4 × Perimeter of the square = 4 × 40

= 160 m

Now,

Side of the doubled perimeter = s’

160 = 4 × s’

s’ = 160/4

s’ = 40 m

Further new area

Area of 4 times perimeter square = s’ × s’

Area of 4 times perimeter square = 40 × 40

Area of 4 times perimeter square = 1600 m²

To find how many times the area of the square increased after 4 times the perimeter

= 

= 1600/100

= 16 times

Therefore,

Area of the square is increased by 16 times when the perimeter of the square is 4 times.

Question 5. Find the perimeter of the square if its area is 400 cm²?

Solution:

Area of square = 400 cm²

side × side = 400

s² = 400

s = √400

s = 20 cm

Side of the square = 20 cm

Perimeter of square = 4 × side

Perimeter of square = 4 × 20

Perimeter of square = 80 cm