Regular pentagon SMILE is fully contained in the interior of regular pentagon FROWN. The edges of FROWN are three times the lengths of the edges of SMILE. The area of the region between the two pentagons is 3. Compute [SMILE]
Answers
Given : Regular pentagon SMILE is fully contained in the interior of regular pentagon FROWN.
The edges of FROWN are three times the lengths of the edges of SMILE. The area of the region between the two pentagons is 3.
To Find : Area of [SMILE]
Solution:
Regular pentagon SMILE is fully contained in the interior of regular pentagon FROWN
Regular pentagon would be similar
Hence ratio of area of pentagon = ( ratio of sides)²
The edges of FROWN are three times the lengths of the edges of SMILE
Ratio of sides/edge of FROWN/SMILE = 3/1
Ratio of area = FROWN/SMILE = (3/1)² = 9/1
area of FROWN = 9 * area of SMILE
area of FROWN - area of SMILE = 3
=> 9 * area of SMILE - area of SMILE = 3
=> 8 * area of SMILE = 3
=> area of SMILE = 3 /8 sq units
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