Two overlapping regular pentagons share two vertices. What fraction is the overlap area?

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## Solution

The gold fraction is 1/(5φ+4), which is approximately 8,27%.

The basic knowledge needed is that a regular pentagon’s diagonal length is φ times its side length, where φ is the Golden Ratio. Scaling such that the gold area is 1, and using also the fact that the diagonals trisect the corners, we easily find that A=φ and B=1.

From triangle similarity and comparing the bases, we find that D=φ^{2}=φ+1. Since C+1=D, we get that C=φ. Adding it all up, we find for the total area 5φ+4.

## Poem

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Just not to be unknown