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Well, to calculate the area of the star it is more easier to calculate the area of the pentagon and to subtract the areas of the small isosceles triangles formed outside the star shape.
Since the pentagon is regular, the base angles of the isosceles triangles measure 36 degrees each. The bases of these isosceles triangles are the sides of pentagon and the measure 10 cm.
To determine area of each isosceles triangles, we'll have to calculate its heights. We'll determine the height from any right angle triangle formed when we draw the height within each isosceles triangles using tangent function.
tan 36 = height (opp.)/(10/2) (adj.)
height = 5*tan 36
height = 3.63 cm
We'll calculate the area of isosceles triangle;
A = base*height/2
A = 10*3.63/2
A = 18.15 ` `
Now, we'll calculate the total area of these isosceles triangles:
A total = 5* 18.15
A total = 90.75 `cm^(2)`
The area of pentagon is formed from the areas of the three triangles fromed when drawing 2 diagonals of pentagon.
To determine th length of diagonal, we'll consider the isosceles triangle, whose equal sides are the sides of pentagon. The top angle is one of the interior angles of pentagon and it measures 108 degrees.
Since we also need to calculate the height of these triangles, we'll draw the height and we'll notice that inside isosceles triangle there are formed two right angle triangles, whose hypotenuses are the sides of pentagon.
We'll use the sine function to determine the height and we'll use cosine fuction to determine the half length of diagonal.
sin 36 = height/10
height = 10*sin 36
height = 5.87
cos 36 = (diagonal/2)/10
diagonal/2 = 10*cos 36
diagonal/2 = 8.09
diagonal = 16.18 cm.
Area of the isosceles triangle is:
A triangle = 5.87*16.18/2
A triangle = 47.48 `cm^(2)`
We'll determine the area of the middle isosceles triangle.
A mid. triangle = diagonal*diagonal*sin 36/2
A mid. triangle = `16.18^(2)`
A mid. triangle = 76.93 `cm^(2)`
The area of pentagon = 2*A triangle + A mid. triangle
Area pentagon = 2*47.48 + 76.93
Area pentagon = 171.89 `cm^(2)`
The area of the star = A pentagon - A total
The area of the star = 171.89 - 90.75
The area of the star = 81.14 `cm^(2)`
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