In the figure A1, is the area bounded by a square and a quarter circle with center a vertex of the square. A2 is the area bounded by the square and the circle. What is the ratio of A1 to A2?

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In the figure A1, is the area bounded by a square and a quarter circle with center a vertex of the square. A2 is the area bounded by the square and the circle. What is the ratio of A1 to A2?

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Answer 1 · 2018-06-20T04:09:29

$\frac{A_{1}}{A_{2}} = \frac{2 π}{π - 2}$ $(A_1)/(A_2)=(2pi)/(pi-2)$

Explanation:

enter image source here
Let $s$ $s$ be the side length of the square.
Let $O and r$ $O and r$ be the center and the radius of the circle, respectively, as shown in the figure.
$\Rightarrow$ $=>$ as $∠ A O B = 90^{\circ}, \Rightarrow s = \sqrt{2} r$ $angleAOB=90^@, => s=sqrt2r$
$A_{1} =$ $A_1=$ Yellow area $= \frac{π s^{2}}{4} = \frac{π {(\sqrt{2} r)}^{2}}{4} = \frac{1}{2} π r^{2}$ $=(pis^2)/4=(pi(sqrt2r)^2)/4=1/2pir^2$
$A_{2} =$ $A_2=$ Green area $=$ $=$ area of sector $O A B -$ $OAB -$ area of $DeltaOAB$
$=(pir^2)/4-r^2/2$
$=((pi-2)r^2)/4$

$(A_1)/(A_2)=((pir^2)/2)/(((pi-2)r^2)/4)=(2pi)/(pi-2)~~5.504$

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In the figure A1, is the area bounded by a square and a quarter circle with center a vertex of the square. A2 is the area bounded by the square and the circle. What is the ratio of A1 to A2?

1 Answer

Explanation:

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