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Geometry Level 3

The diagram below shows a smaller pentagon formed inside a large, convex pentagon by joining each vertex of the large pentagon with its non-adjacent vertices. Each of the 5 colored triangles has the same area of 1. Find the ratio of the area of the large pentagon to the area of the smaller pentagon.

If the ratio can be expressed as \[\dfrac{a\sqrt{b} + c}{d},\] where \(a,b,c,d\) are positive integers, \(b\) is square-free, input the smallest possible value of \(a + b + c + d\) as your answer.

Diagram not drawn to scale

Diagram not drawn to scale


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