Find the probability that a point chosen at random in the figure lies in the shaded region. Figure: http://www.flickr.com/photos/93084714@N07/8624622629/
The figure is that of a regular pentagon within a circle with radius 4. The area of the circle is pi*16. The area of the pentagon is equal to that of 5 isosceles triangles with the equal sides equal to 4 and the enclosed angle equal to 72 degrees. A perpendicular dropped from the center bisects the side of the pentagon. Let the length of the base of the triangle be 2x and the height of the triangle is y.
`sin 36 = x/4`
=> `x = 4*sin 36`
`cos 36 = y/4`
=> `y = 4*cos 36`
The area of each triangle is `(1/2)*8*sin 36*4*cos 36` . The area of the pentagon is `80*sin 36*cos 36 ~~ 38.04` .
This gives the area of the shaded area as `pi*16 - 38.04 ~~ 12.22`
The probability that a point chosen at random lies in the shaded region is `12.22/50.2654 ~~ 24.31%`
The required probability is 24.31%