Solution to Mensa Math Brain Buster

 

Let r be the length of the radius of each circle. Then the area of each circle inside and outside the polygon is:  π r² .

A2 = the area  of the 17 circles minus the area of all circular sectors inside the polygon, i.e.

A2 =  17 π r²   -  A1

The internal angles of an n-gon (polygon of n sides) total: 

(n - 2) π radians.  (This is given that an n-gon can be divided into (n-2) triangles and the sum of the angles of a triangle =  π  radians (180 deg)

The particular polygon has 17 sides, so its internal angles total: 

(17 - 2) π   =   15 π  radians 

The area of the circular sectors inside the polygon is equal to the area of all 17 circles multiplied by the proportion that is inside the polygon.

A1  =17 π r² ·  15 π  / ( 17 · 2π ) =  π r² · 15/2

=  (15/2) π r²

A2 =  17 π r²   -  A1 =  17 π r²   - ( π r² · 15/2) = 

π r² (17- 15/2) =  (19/2) π r² 

A2 - A1 =  (19/2) π r²  -   (15/2) π r²  =  (4/2) π r²

=  2 π r²