SOLUTION: An equilateral triangle is inscribed in a circle with a radius of 6". Find the area of the segment cut off by one side of the triangle

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Question 254908: An equilateral triangle is inscribed in a circle with a radius of 6". Find the area of the segment cut off by one side of the triangle
Answer by edjones(8007)   (Show Source): You can put this solution on YOUR website!
Solved by pluggable solver: Calculate side length of equilateral triangle inscribed in the circle

The circle of radius 6 with center O with an equilateral triangle ABC inscribed in it.


Join edges A,B and C with center O as shown in figure.





Consider Triangle AOB and AOC,


1.> .............(Sides of Equilateral triangle are equal)


2.> .............(These are radius of a circle)


3.>OA is common side to both Triangles.


From conditions 1,2 and 3


Triangle AOB and AOC are congruent to each other. (SSS congruency condition)


Now since triangles are congruent,


Therefore, ...............(4)


Similarly, ...............(5)


As, Angle CAB is an angle in a equilateral triangle


Hence, degrees ...............(6)


Now,


..........(From (4))





degrees


degrees ...........(6)


Similarly,


and from condition 5, degrees ...........(7)


Now consider Triangle AOB,


Sum of angles in a Triangle is 180 degrees.


Hence,


........(From 6 and 7)





...........(8)


In Triangle AOB using sine rule,




















Hence the side of an equilateral triangle inscribed in a circle of radius 6 is 10.3923048.


.
A(equilateral triangle)=s^2sqrt(3)/4
(10.39^2*sqrt(3))/4
=46.74 sq in
.
A(circle)=pi*r^2
=pi*6^2
=113.1 sq in
.
113.1-46.74=66.35 area of all 3 segments cut off by a side of the triangle.
66.35/3
=22.12 sq in, area of the segment cut off by one side of the triangle.
.
Ed
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