Question 832764
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Hi, there--

THE PROBLEM:
A circle is inscribed in a square which is in turn inscribed of a larger circle. what is the ratio of the area of 
the larger circle to that of the smaller circle?

A SOLUTION:
You want the ratio of the area of the larger circle to that of the smaller circle.

Sandwiched between these two circles is a square. (See drawing below.)
We say that the smaller circle is inscribed in the square.
Likewise, the square is inscribed in the larger circle.

{{{
drawing( 400, 300, -10, 10, -10, 10,
  circle( 0, 0, 2.5 ),
  circle( 0, 0, 2.5*sqrt(2)),
  rectangle( -2.5, -3.3, 2.5, 3.15),
  locate( 0, 0, O),
  locate( 0, 4.3, A),
  locate( 2.9, 4.2, B),  
  locate(-3, 0.2, s),

  line( 0, 0, 0, 3.3),
  line( 0, 0, 2.5, 3.15)
  )
}}}


AREA OF SMALL CIRCLE
In order to find the area of a circle, we need to know its radius. Let's begin with the smaller circle. Do you
see that the radius of the smaller circle (AO) is one-half the side length of the square?

Let s be the side length of the square.
Then s/2 is the radius of the smaller circle.

An expression for the area of the smaller circle is {{{pi*r^2=pi*(s/2)^2}}} or {{{(s^2pi/4)}}}.

AREA OF LARGE CIRCLE
Now consider the larger circle. Do you see that the radius of this circle (OB) is one-half the diameter
of the square?

To find the diameter of the square, we use the Pythagorean Equation, {{{a^2+b^2=c^2}}}.
The length and width of the square are the legs of a right triangle, the diagonal is the hypotenuse.

{{{c^2=s^2+s^2}}}
{{{c^2=2s^2}}}

To find c, we take the square root of both sides of the equation.
{{{c=sqrt(2s^2)}}}

Simplify. (The square root of c^2 is c, and the square root of s^2 is s.)
{{{c=s*sqrt(2)}}}

The length of the diagonal of the square is {{{s*sqrt(2)}}}. The radius of the larger circle is half the length
of the diagonal, or {{{(s*sqrt(2))/2}}}.

An expression for the area of the larger circle is {{{pi*r^2=pi*((s*sqrt(2))/2)^2}}}.

We can simplify this expression. (The (sqrt(2))/2 squared is 2/4=1/2.)
{{{pi*((s*sqrt(2))/2)^2=(pi*s^2)/2}}}

RATIO OF AREAS
Let R be ratio of the area of the larger circle to the area of the smaller circle.

{{{R=((s^2*pi)/2)/((s^2*pi)/4)}}}

Simplify. The s^2 terms and pi cancel out.
{{{R=((cross(s^2*pi))/2)/((cross(s^2*pi))/4)}}}
{{{R=(1/2)/(1/4)}}}
{{{R=2}}}

The ratio of the area of the larger circle to the area of the smaller one is 2:1. In other words the larger 
circle has twice the area of the smaller one. Furthermore, since we use s for the side length of the square, 
we know that this ratio is true for a square of any side length.


Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com
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