## PERIMETER OF RECTANGLE

A rectangle is a four-sided closed figure where the lengths of opposite sides will be equal and each vertex angle will be right angle or 90o as shown below.

## Formula for Perimeter of Rectangle

Let l be the length and w be the width of a rectangle.

Then, the formula for perimeter of the rectangle :

Perimeter  =  2(l + w)

## Examples

Example 1:

The length and width of a rectangle are 16 cm and 12 cm respectively. Find its perimeter.

Solution:

Formula for perimeter of a rectangle :

=  2(l + w)

Substitute 16 for l and 12 for w.

=  2(16 + 12)

=  2(28)

=  56

So, the perimeter of the rectangle is 56 cm.

Example 2:

If the perimeter of a rectangle is 50 cm and its length is 15 cm, then find its width.

Solution:

Perimeter of the rectangle  =  50 cm

2(l + w)  =  50

Divide each side by 2.

l + w  =  25

Substitute 15 for l.

15 + w  =  25

Subtract 15 from each side.

w  =  10

So, the width of the rectangle is 10 cm.

Example 3 :

The area of the rectangle is 150 square inches. If the length is twice the width, then find its perimeter.

Solution:

Let x be the width of the rectangle.

Then, the length of the rectangle is 2x.

Area of the rectangle  =  150 in2

⋅ w  =  150

⋅ 2x  =  150

2x2  =  150

Divide each side by 2.

x2  =  75

Find positive square root on both sides.

√x =  √75

x  =  √(5 ⋅ 5 ⋅ 3)

x  =  5√3

Therefor, the width of the rectangle is 5√3 in.

Then, the length of the rectangle is

=  2 ⋅ width

=  2 ⋅ 5√3

=  10√3 in

Formula for perimeter of a rectangle :

=  2(l + w)

Substitute 10√3 for l and 5√3 for w.

=  2(10√3 + 5√3)

=  2(15√3)

=  30√3

So, the perimeter of the rectangle is 30√3 in.

Example 4:

The length of a rectangle is 3 ft and one of the diagonal measures √13 ft. Find its perimeter.

Solution:

To find the perimeter of a rectangle, we have to know its length and width. Length is given in the question, that is 3 ft. So, find its width.

Draw a sketch.

In the figure shown above, consider the right triangle ABC.

By Pythagorean Theorem, we have

AB2 + BC2  =  AC2

Substitute.

AB2 + 32  =  (√13)2

Simplify and solve for AB.

AB2 + 9  =  13

Subtract 9 from each side.

AB2  =  4

Find positive square root on both sides.

√AB2  =  √4

AB  =  2

Therefore, the width of the rectangle is 2 ft.

Formula for perimeter of a rectangle.

=  2(l + w)

Substitute 3 for l and 2 for w.

=  2(3 + 2)

=  2(5)

=  10

So, the perimeter of the rectangle is 10 ft.

Example 5:

The length of a rectangle is 3 yards more than its width and its perimeter is 18 yards. Find its length and width.

Solution:

Let x be the width of the rectangle.

Then, the length of the rectangle is (x + 3) yards.

Perimeter of the rectangle  =  18 yards

2(l + w)  =  18

Divide each side by 2.

l + w  =  9

Substitute (x + 3) for l and x for w.

(x + 3) + x  =  9

x + 3 + x  =  9

2x + 3  =  9

Subtract 3 from each side.

2x  =  6

Divide each side by 2.

x  =  3

x + 3  =  6

So, the length of width of the rectangle are 6 yards and 3 yards respectively.

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