Shown below is a regular pentagon inscribed in a circle. Calculate the area of the shaded region. Round your answer to the nearest tenth.

S(a) = 27,5036  squared units

Step-by-step explanation:

Shaded area is :

S(a)  =  Area of the circle  - area of the regular pentagon      (1)

A(c) = area of the circle

A(c) = π*(r)²    ⇒  A(c) =  π*(6)²     ⇒   A(c) =  36*π   ⇒ A(c) = 113,0976 squared units

Area of a regular pentagon:

a) If we draw a straight line between the center and each vertex we get  5 triangles, and if we draw the apothem for each side, we get 10 triangles. We will calculate the area of one  of these triangles

The first 5 triangles has a central angle equal to 72⁰ according to:

360/5  =  72

When we divide these triangles in two triangles by means of the apothem, each central angle will be of 36⁰, then

sin 36⁰  =  0,58778  and cos 36⁰ = 0,809017    and   sin 36⁰ =  x/6      here x is half of the side of the regular pentagon. Then

0,58778  =  x/6

x = 6*0,58778

x =  3,52668 units of length  

and cos 36⁰ = a/6      where a is the apothem, then

0,809017  = a / 6     ⇒   a  = 6*0,809017

a =  4,8541 units of length

Now we are in conditon to calculate area of the triangles as:

A(t) = (1/2)*b*h

A(t) = (1/2)*x*a     ⇒ A(t) = 0,5* 3,52668*4,8541

A(t) = 8,5594 squared units

Finally we have 10 of these triangles, then

Area of regular pentagon is : 10*A(t) squared units

A(p) = 85,594 squared units

Now plugging these values in equation (1) we get the shaded area

S(a) =  113,0976 - 85,594

S(a) = 27,5036  squared units

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step-by-step explanation: