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About 103.2 cm^2
To find the area of the shaded region, we will draw segments from the remaining vertices of the pentagon to the pentagon's center, which creates 5 congruent triangles. We know they are congruent by the SSS (Side-Side-Side) Congruence Theorem.
The sum of the interior angles is 540^(∘). Since this is a regular pentagon, each interior angle will measure 540^(∘)5=108^(∘). Additionally, the segment between each vertex and the center of the pentagon bisects the angles at each vertex. Therefore, if each of the pentagon's angles is 108^(∘) half of this is 54^(∘).
If we draw the height from the vertex angle, it cuts the opposite side in two equal halves.
Substitute values
LHS * 5=RHS* 5
Rearrange equation
Use a calculator
Round to 2 decimal place(s)