Question 320076: A circle is inscribed in a regular pentagon. The perimeter of the pentagon is 150 cm. Find the area of the circle.
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! A circle is inscribed in a regular pentagon. The perimeter of the pentagon is 150 cm. Find the area of the circle.
a regular pentagon has 5 equal sides,
the circle inscribed in the pentagon touches at each of the midpoints of these 5 sides,
perimeter of the pentagon is 150 cm,
150/5 = 30 so each side of the pentagon is 30 cm,
there are 5 equal triangles inside the pentagon each having angles of 72 and 54 and 54 (180 - 72 = 108, 108/2 = 54),
the 72 degree angle of one of the triangles is opposite a side length of 30 cm,
1/2 of each of the triangles is a 36-54-90 triangle (72/2 = 36),
the side opposite the 36 degree angle is 15 cm (30/2 = 15),
sin = opposite (opp) / hypotenuse (hyp),
sin 36 = 15/hyp
hyp * sin 36 = 15
hyp = 15/(sin 36)
hyp = 25.52 cm approximately (approx.) this is also the radius of the circle,
area of circle = pi * radius squared
area of circle = pi * (25.52)^2 = approx. 2045.95 square cm
area of circle = approx. 2046 square cm
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