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The above figure shows a regular hexagram inscribed in a circle of radius 10 units. Find the area of the shaded region

 

 Jun 17, 2020
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The area of the circle is pi·radius2  =  100 pi

 

If you were to connect the center of the circle to each of the vertices of the interior hexagon, you would

get 6 congruent equilateral triangle and each of these triangles would be congruent to the six 

equilateral triangles "at the points".

 

A radius to one of the points on the circle = 10, so the height of each of the equilateral triangles is 5.

This means that the base of the triangle is ( 5 / sqrt(3) ) · 2  =  10/sqrt(3)

The area of each triangle is  ½ · ( 10/sqrt(3) ) · 5  =  25 / sqrt(3).

Since there are 12 congruent triangles, the total area is 300/sqrt(3)  =  100·sqrt(3)

 

The area inside the circle but outside the hexagram =  100 pi - 100·sqrt(3)

Of this area, one-third is shaded  =  ( 100 pi - 100 sqrt(3) ) / 3  

 Jun 17, 2020

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