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Circle $1$ is circumscribed about a square of side $8$ and circle $2$ is inscribed in the square. What is the ratio of the area of circle $1$ to circle $2$?

I know the formula of a circle is $A=\pi r^2$ but does anyone know how I can find the radius of circle $1$ and circle $2$?

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  • $\begingroup$ What points on the square does each circle touch? $\endgroup$ – John Douma Mar 20 '16 at 17:58
  • $\begingroup$ the question doesn't say but i assume the vertices $\endgroup$ – kero Mar 20 '16 at 17:59
  • $\begingroup$ Draw it. You will see what I mean. $\endgroup$ – John Douma Mar 20 '16 at 18:00
  • $\begingroup$ Circle 1 is circumscribed so the vertices of the square are in the circle. So what is the radius of the circle 1 from the center to a vertex in terms of the dimensions of the square? (There is one straightforward correct answer). Circle 2 is inscribed inside the square. What are the points of the square where circle 2 touches it. (The points are not the vertices. Draw it and you will see.) What are these points and what is the radius of circle 2 that goes from the center to these points. You now know the radii. So you can get the areas. $\endgroup$ – fleablood Mar 20 '16 at 18:33
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$C_2$ is the circumcircle of a square $Q_2$ having one half of the area of the original square $Q_1$. Therefore $${{\rm area}(C_1)\over {\rm area}(C_2)}={{\rm area}(Q_1)\over {\rm area}(Q_2)}=2\ .$$

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The circle inscribed in the square has a radious $r_1=4$, because it touches the sides of the square at the middle points. While the circle ciscunscribed about the square has a radious of $r_2=4\sqrt{2}$ becouse it touches the vertex of the square (its radious is just half of the diagonal of the square). Finally, the ratio between the areas of both circles is just the difference \begin{equation} A_2/A_1=\frac{\pi r_{2}^2}{\pi r_{1}^2}=2 \end{equation}

inscribed and circunscribed circles about an square

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  • $\begingroup$ I believe the problem is asking for the ratio of the areas, not the difference between them. $\endgroup$ – John Douma Mar 20 '16 at 18:12
  • $\begingroup$ .... which makes the square having side 8 irrelevent. Any square will result in the same ratio. The ratio of the radii are $\sqrt{2}:1$ so the ratio of the areas will be $2:1$. $\endgroup$ – fleablood Mar 20 '16 at 18:15
  • $\begingroup$ Actually you are right! I have changed the solution $\endgroup$ – seoanes Mar 20 '16 at 18:17
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The circle diameter has become distance between corners of a square.Size has grown by a factor or ratio $ \sqrt 2 $ and area by its square, viz $ {(\sqrt 2)}^2 =2. $ This can be continued in a geometrically expanding series as well.

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"does anyone know how i can find the radius of circle 1 and circle 2?"

Circle 1. The vertices of the square are in the circle. That's what "circumscribe" means. The diagonals of a square bisect each other so the distance from the center of the square to each of the vertices are equal distance. That's four points on a circle each equal distance from a central point.

A little bit of futzing and one can prove given three or more points on a circle there is only one point (namely the center of the circle) and only one distance (the radius) that is equal distance from all the three or more points on the circle.

So the radius of Circle 1 is half the diagonal of the square.

Circle 2. Each side of the square is tangent to smaller circle 2. A bit of futzing and one can prove the perpendiculars of tangents to a circle all intersect in the center of the circle. So the four perpendiculars intersect at the center of the circle. As opposite sides of a square are parrallel it follows these tangents must occur in the midpoints of the sides (otherwise the perpendiculars from the opposite sides wouldn't intersect.)

So the radius of Circle 2 are the lines from the center of the square to the midpoints of the sides of the square.

So the radius of Circle 2 is half the side the square.

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Their radius are $4$ and $1$ respectively. So the ratio should be $$\frac{4^2}{1^2} = 16$$

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  • $\begingroup$ How do you figure the radii are 4 and 1? I get that they are $4\sqrt{2}$ and 4 respectively. $\endgroup$ – fleablood Mar 20 '16 at 18:17

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