In this section, you will learn how to find perimeter and area of square.
A square is a four-sided closed figure where the lengths of all the four sides will be equal and each vertex angle will be right angle or 90^{o }as shown below.
The distance around a two dimensional shape is called perimeter.
If s be the length of each side of a square, then the perimeter of the square is
= s + s + s + s
= 4s
So, the formula for perimeter of a square :
Perimeter = 4s
The amount of space available inside the boundary of a two-dimensional space is called area.
We can use the following two formulas to calculate the amount of space available inside the square.
If s be the length of each side of a square, then the formula for area of a square :
Area = s^{2}
If d be the length of each diagonal of a square, then, the formula for area of a square :
Area = 1/2 ⋅ d^{2}
Example 1 :
If the length of each side of a square is 8.5 cm, then find its perimeter.
Solution :
Formula for perimeter of a square :
= 4s^{ }
Substitute 14 for s.
= 4(8.5)
= 34
So, the perimeter of the square is 34 cm.
Example 2 :
The length of each diagonal of a square is 2√2 cm. Find its perimeter.
Solution :
To find the perimeter of a square, first we have to know the length of each side.
Let s be the length of each side of the square.
Draw a sketch.
In the figure shown below, consider the right triangle ABC.
By Pythagorean Theorem, we have
AB^{2} + BC^{2} = AC^{2}
Substitute.
s^{2} + s^{2} = (2√2)^{2}
Simplify and solve for s.
2s^{2} = 2^{2} ⋅(√2)^{2}
2s^{2} = 4 ⋅(2)
2s^{2} = 8
Divide each side by 2.
s^{2} = 4
Find positive square root on both sides.
√s^{2} = √4
√s^{2} = √(2 ⋅ 2)
s = 2
Formula for perimeter of a square.
Perimeter = 4s
Substitute 2 for s.
= 4(2)
= 8
So, the perimeter of the the square is 8 cm.
Example 3 :
If a square has the side length of 7.5 cm, then find its area.
Solution :
When the length of a side is given, formula for area of a square :
= s^{2 }
Substitute 24 for s.
= (7.5)^{2}
= 56.25
So, area of the square is 56.25 square cm.
Example 4 :
The area of a square is 32 square inches. Find the length of its diagonal.
Solution :
Area of the square = 32 in^{2}
1/2 ⋅ d^{2} = 32
Multiply each side by 2.
d^{2} = 64
Find positive square root on both sides.
√d^{2} = √(8 ⋅ 8)
d = 8
So, the length of diagonal is 8 inches.
Example 5 :
The square has side length 36 inches. Find its area in square feet.
Solution :
When the length of a side is given, formula for area of a square :
= s^{2 }
Substitute 12 for s.
= 36^{2}
= 1296 in^{2} -----(1)
We know
12 inches = 1 ft
Square both sides.
(12 inches)^{2} = (1 ft)^{2}
12^{2} in^{2} = 1^{2} ft^{2}
144 in^{2} = 1 ft^{2}
Therefore, to convert square inches into meter square feet, we have to divide by 144.
(1)-----> Area of the square = 1296 in^{2}
Divide the right side by 144 to convert in^{2} into ft^{2}.
Area of the square = (1296 / 144) ft^{2}
= 9 ft^{2}
So, the area of the square is 9 square feet.
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