# Puzzle

• Jul 7th 2010, 09:43 PM
Chris11
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 $36-18\pi$
• Jul 7th 2010, 10:28 PM
earboth
Quote:

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 $36-18\pi$

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

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

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

$36- 72\left(1-\frac14 \pi\right) = -36+18\pi$
• Jul 8th 2010, 07:55 AM
Soroban
Hello, Chris11!

Another approach . . .
Code:

    D * - - - - - * - * C       |    *:::::::* |       |  *:::::::*  |       |  *::::::*    |       | ::::::*      | 6       |*::::*        |       |:::*          |       *:*            |     A * - - - - - - - * B               6

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

Hence: . $9\pi - 18 \;=\;9(\pi-2)$

Therefore, the lens-shaped region has area:

. . . . $2\cdot9(\pi - 2) \:=\:18(\pi - 2)$
• Jul 8th 2010, 10:01 AM
Chris11
.