Ratio

**reiward** · December 12th 2009, 05:02 AM

We had this in a quiz, my answer was $\sqrt{2} = 1$

but my friends say the answer is 2 = 1 because they squared both sides. but I don't get why the need to square it. so here's the problem.

A square is inscribed in a circle, and another circle is inscribed in the square. What is the ratio of the area of the larger circle to the small circle?

What I did was

Let A1 = large circle
Let A2 = small circle

Radius of large circle:
$R = \frac {\sqrt{2}}{2} e$

Radius of small circle:
$r = \frac {1}{2} e$

$A1 = A2$

$\pi R^2 = \pi r^2$

I then cancel out pi and square leaving the radii.

$\frac {\sqrt{2}}{2} e= \frac {1}{2} e ] 2$

Thus becomes $\sqrt{2} = 1$

**Krahl** · December 12th 2009, 05:26 AM

If you find a ratio you cant square it.
for example say im 3 times your height. so ratio is 3:1. square it 9:1 so now im 9 times your height.
But i find that 2:1 is the correct answer. If your friends got this answer by squaring your answer then they got it through the wrong methods

Draw a circle, then draw a square within that circle then draw a circle within that square.

the radius R of the large circle is from the centre of the small circle to the corner of the square.

the radius r of the small circle is from the centre of the small circle to the side of the square. but then from that side of the square to the edge of the square should also be r.(since half the side of the square is r).

so we get a right angled triangle with hypoteneuse R and sides r.

so $R^2=r^2+r^2=2r^2$

$A1=(pi) R^2=2(pi)r^2$
$A2=(pi)r^2$

so ratio of 2:1

**Soroban** · December 12th 2009, 12:52 PM

Hello, reiward!

Sorry, your friends are right . . .

A square is inscribed in a circle, and another circle is inscribed in the square.
What is the ratio of the area of the larger circle to the small circle?

Consider one quadrant of the diagram.

Code:

      *
      |     *
      |         *   B
    A * * - - - - *
      |     *   * | *
      |       *   |
      |     *   * |   *
      |   *       |    *
      | *         *
    O * - - - - - * - - *
                  C

The radius of the large circle is: . $OB = R$
The area of the large circle is: . $A_1 \,=\,\pi R^2$

Since $\Delta OBC$ is a 45-45-90 right triangle: . $BC \,=\,\frac{R}{\sqrt{2}}$
The radius of the small circle is: . $r \,=\,\frac{R}{\sqrt{2}}$
The area of the small circle is: . $A_2 \:=\:\pi r^2 \:=\:\pi\left(\frac{R}{\sqrt{2}}\right)^2 \:=\:\frac{\pi R^2}{2}$

The ratio of areas is: . $\frac{A_1}{A_2} \;=\;\frac{\pi R^2}{\frac{\pi R^2}{2}} \;=\;2 \quad\Rightarrow\quad 2:1$

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