1. Puzzle

What is wrong with the following 'solution' to the following problem?

Let ABCD be a square with side lengh of six. Circular arcs of radius six are drawn with centers at B and D. Find the area of the intersection of the areas in the square bounded by the arcs.

The area of the region of the square that is above the oval shaped intersection is 36-1/4 the area of one of the circle. Now there are 2 of these. Hence, the area of the oval shaped intersection is $\displaystyle 36-18\pi$

2. Originally Posted by Chris11
What is wrong with the following 'solution' to the following problem?

Let ABCD be a square with side lengh of six. Circular arcs of radius six are drawn with centers at B and D. Find the area of the intersection of the areas in the square bounded by the arcs.

The area of the region of the square that is above the oval shaped intersection is 36-1/4 the area of one of the circle. Now there are 2 of these. Hence, the area of the oval shaped intersection is $\displaystyle 36-18\pi$

The area which doesn't belong to the red "2-gon" is calculated by:

$\displaystyle 2\left(36- \frac14 \cdot \pi \cdot 6^2\right) = 72\left(1-\frac14 \pi\right)$

2. Now the area of the red "2-gon" is calculated by:

$\displaystyle 36- 72\left(1-\frac14 \pi\right) = -36+18\pi$

3. Hello, Chris11!

Another approach . . .
Code:
    D * - - - - - * - * C
|     *:::::::* |
|   *:::::::*   |
|  *::::::*     |
| ::::::*       | 6
|*::::*         |
|:::*           |
*:*             |
A * - - - - - - - * B
6

. . $\displaystyle \begin{array}{cccccc} \text{a quarter-circle:} & \frac{1}{4}\pi(6^2) &=& 9\pi \\ \text{minus} \\ \text{half the square:} & \frac{1}{2}(6^2) &=& 18 \end{array}$
Hence: .$\displaystyle 9\pi - 18 \;=\;9(\pi-2)$
. . . . $\displaystyle 2\cdot9(\pi - 2) \:=\:18(\pi - 2)$