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 90^{o }as shown below.
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)
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 in^{2}
l ⋅ w = 150
x ⋅ 2x = 150
2x^{2} = 150
Divide each side by 2.
x^{2} = 75
Find positive square root on both sides.
√x^{2 } = √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
AB^{2} + BC^{2} = AC^{2}
Substitute.
AB^{2} + 3^{2} = (√13)^{2}
Simplify and solve for AB.
AB^{2} + 9 = 13
Subtract 9 from each side.
AB^{2} = 4
Find positive square root on both sides.
√AB^{2} = √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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