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.
Let s be the length of each side of a square.
Then, the formula for area of a square :
Area = s^{2}
Let 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 :
Find the area of the square having side length 24 cm.
Solution :
When the length of a side is given, formula for area of a square :
= s^{2}
Substitute 24 for s.
= 24^{2}
= 576
So, area of the square is 576 square cm.
Example 2 :
If the area of a square is 64 square inches, then find the length of each side.
Solution :
Area of the square = 64 in^{2}
s^{2} = 64
Find positive square root on both sides.
√s^{2} = √64
s = 8
So, the length of each side of the square is 8 inches.
Example 3 :
The square has side length 250 cm. Find its area in square meter.
Solution :
When the length of a side is given, formula for area of a square :
= s^{2}
Substitute 250 for s.
= 250^{2}
= 62500 cm^{2} ----(1)
We know
100 cm = 1 m
Square both sides.
(100 cm)^{2} = (1 m)^{2}
100^{2} cm^{2} = 1^{2} m^{2}
10000 cm^{2} = 1 m^{2}
Therefore, to convert centimeter square into meter square, we have to divide by 10000.
(1)----> Area of the square = 62500 cm^{2}
Divide the right side by 10000 to convert cm^{2} into m^{2}.
Area of the square = (62500/10000) m^{2}
= 6.25 m^{2}
So, the area of the square is 6.25 square meter.
Example 4 :
If the length of each diagonal is 2√2 cm, then find its area.
Solution :
When the length of a diagonal is given, formula for area of a square :
= 1/2 ⋅ d^{2}
Substitute 2√2 for d.
= 1/2 ⋅ (2√2)^{2}
Simplify.
= 1/2 ⋅ (4 ⋅ 2)
= 1/2 ⋅ (8)
= 4
So, the area of the square is 4 square cm.
Example 5 :
If the lengths of the diagonals of two squares are in the ratio 2 : 5. then find the ratio of their areas.
Solution :
From the ratio 2 : 5, let the diagonals of two squares be 2x and 5x respectively.
When the length of a diagonal is given, formula for area of a square :
= 1/2 ⋅ d^{2}
Area of 1^{st} square = 1/2 ⋅ (2x)^{2} = 1/2 ⋅ (4x^{2}) = 4x^{2} / 2 |
Area of 2^{nd} square = 1/2 ⋅ (5x)^{2} = 1/2 ⋅ (25x^{2}) = 25x^{2} / 2 |
Ratio of the areas :
= (4x^{2} / 2) : (25x^{2} / 2)
Multiply each term of the ratio by 2.
= 4x^{2} : 25x^{2}
Divide each term by x^{2}.
= 4 : 25
So, the ratio of the areas of two squares is 4 : 25.
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