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By adding point O at the center of the inner pentagon, you can make five congruent isosceles triangles. The angle at point O is 72°, so these are 72-54-54 triangles.
The triangles outside the inner pentagon that make up the five points of the star are 72-72-36 triangles. You get this because you know the inner angles for a regular pentagon are 108°.
From here, just take advantage of knowing that the area of the five outside triangles to the area of the five inside triangles have to be in the ratio tan 72° to tan 54°. So, the five outside triangles have area = tan 72° / tan 54° = 3.0777 / 1.3764 = 2.236. Therefore, the area of the star is 2.236 + 1 = 3.236.
By the way, you didn't say you wanted an exact answer (using radicals), but the exact answer is interesting: tan 72° / tan 54° = √5. So, the area of the five outside triangles is exactly √5, and the area of the star is exactly √5 + 1.
The triangles outside the inner pentagon that make up the five points of the star are 72-72-36 triangles. You get this because you know the inner angles for a regular pentagon are 108°.
From here, just take advantage of knowing that the area of the five outside triangles to the area of the five inside triangles have to be in the ratio tan 72° to tan 54°. So, the five outside triangles have area = tan 72° / tan 54° = 3.0777 / 1.3764 = 2.236. Therefore, the area of the star is 2.236 + 1 = 3.236.
By the way, you didn't say you wanted an exact answer (using radicals), but the exact answer is interesting: tan 72° / tan 54° = √5. So, the area of the five outside triangles is exactly √5, and the area of the star is exactly √5 + 1.