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
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