A square is inscribed in a circle of radius 1 unit, and a larger square is circumscribed about the same circle. What is the area of the region located between the two squares?

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A square is inscribed in a circle of radius 1 unit, and a larger square is circumscribed about the same circle. What is the area of the region located between the two squares?

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Answer 1 · 2018-01-11T02:33:33

$A = 2 {unit}^{2}$ $A=2 " unit"^2$

Explanation:

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Area of a square is given by : $A = \frac{1}{2} d^{2}$ $A=1/2d^2$ , where $d$ $d$ is the length of the diagonal of the square.
Given radius of the circle $r = 1$ $r=1$ ,
$\Rightarrow O A = 1$ $=> OA=1$ ,
$\Rightarrow diagonal of the smaller square = A B = 2 \cdot O A = 2$ $=> "diagonal of the smaller square"=AB=2*OA=2$ ,
$\Rightarrow area of the smaller square A_{S} = \frac{1}{2} \cdot A B^{2} = \frac{1}{2} \cdot 2^{2} = 2$ $=> "area of the smaller square " A_S=1/2*AB^2=1/2*2^2=2$
$\Rightarrow O E = radius = 1, \Rightarrow E D = 1$ $=> OE="radius"=1, => ED=1$ ,
$\Rightarrow C D = 2 E D = 2$ $=> CD=2ED=2$
$\Rightarrow area of the larger square A_{L} = C D^{2} = 2^{2} = 4$ $=> "area of the larger square " A_L= CD^2=2^2=4$
Hence, area of the region located between the two squares $=$ $=$ shaded area $= A_{L} - A_{S} = 4 - 2 = 2 {unit}^{2}$ $= A_L-A_S=4-2=2 " unit"^2$

Answer 2 · 2018-01-11T04:23:27

Difference in areas of the two squares = 2 sq. units

Explanation:

radius of circle $r = 1$ $r = 1$

Diagonal of inner square $d_{s} = 2 r = 2$ $d_s = 2r = 2$

Area of inner square $A_{s} = (\frac{1}{2}) {(d_{s})}_{s}^{2} = (\frac{1}{2}) \cdot 2^{2} = 2$ $A_s = (1/2) (d_s)^2 = (1/2) * 2^2 = 2$

Side of outer square $a_{S} = 2 r = 2$ $a_S = 2r = 2$

Area of outer square $A_{S} = a^{2} = 2^{2} = 4$ $A_S = a^2 = 2^2 = 4$

Difference in areas = 4 - 2 = 2 sq. units

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A square is inscribed in a circle of radius 1 unit, and a larger square is circumscribed about the same circle. What is the area of the region located between the two squares?

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