SOLUTION: In a circle with diameter of 20 cm, a regular five-pointed star touching its circumference is inscribed. Find the area of the star.

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Question 1180960: In a circle with diameter of 20 cm, a regular five-pointed star touching its circumference is inscribed. Find the area of the star.
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!

there are central angles
each angle is degrees
label a point of the star on the circumference A, center of the circle O and the intersection of the side of A and the side of the point of the star adjacent to A going clockwise B.
angle AOB is degrees and angle AOB is degrees
for triangle AOB, we have

degrees
radius of circle = cm
Now use the law of sines



Area of triangle
Area of the five pointed star =


Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!


By sketching in a few other lines, it shouldn't take you long
to figure out that the star is made up of 10 triangles
all congruent to ΔOPQ.  Also, you can easily get:

∠POQ = 36o, OQ = 20, ∠OQP = 18o, ∠OPQ = 126o

By the law of sines,





Solve that and get

PQ = 14.53085056

Then to find the area of that triangle, we use the
formula:







We multiply that by 10 and get

449.0279766 cm2    <---answer

Edwin
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