# If the side of a square is doubled, then find its perimeter

• Last Updated : 20 Nov, 2021

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)

### If the side of a square is doubled, then find its perimeter.

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,

The side of the square is doubled.

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

Therefore,

s’ = 2s

We know,

P’ = 4s’

P’ = 4 x (2s)

P’ = 8s

Thus,

Perimeter becomes 8 times the original perimeter.

### Sample Questions

Question 1. If the perimeter of the square is 28 cm, calculate how much its sides have to be increased to make its perimeter double?

Solution:

Here we have to find the sides of the square to be increased to make its perimeter double.

Given:

Perimeter of square is 28 cm

As we know that

Perimeter of square = 4 x side (s)

28 = 4 x s

s = 28/4

s = 7 cm

When the perimeter of square is doubled

2 x  perimeter of square = 2 x 28 = 56 cm

Further finding how much times the side has to be increased

As we know that

Perimeter of square = 4 x side (s’)

56 = 4 x s’

s’ = 56/4

s’ = 14 cm

Now,

How much times the side of the square has to be increased to make its perimeter double

= s’/s

= 14/7

= 2 times

Therefore,

The side of the square has to be increased twice in order to make its perimeter double.

Question 2. Derive the general formula when the side of the square is increased by n times.

Solution:

Since, perimeter of a square is equivalent to 4 x side.

Originally,

P = 4 x s

Now, side is made n times.

Therefore,

s’ = n x s

New perimeter, P’ = 4 x s’

= 4 x n x s

= n P

Question 3. Using the above formula, find the perimeter when the side of the square is made 1/3 times.

Solution:

Using the above-derived formula,

P’ = n P

Therefore,

Perimeter of the square becomes 1/3 times.

Question 4: What happens to the perimeter of a square if only two sides are doubled?

Solution:

Since,

Perimeter is given by, P = s + s + s + s

Since, two sides are doubled, we have,

P’ = s + s + 2s + 2s

P’ = 2s + 2s + 2s

P’ = 6s

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