SOLUTION: In a circle of radius 4 cm, find the area of the segment bounded by an arc of measure 120 degrees and the chord joining the endpoint of the arc. Round your answer to the nearest te

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Question 333857: In a circle of radius 4 cm, find the area of the segment bounded by an arc of measure 120 degrees and the chord joining the endpoint of the arc. Round your answer to the nearest tenth.
Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
the area of the circle wedge created by the arc with central angle of 120
is 120/360 = 1/3 of the area of the circle
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Area of arc = 1/3*pi%2A4%5E2=16*pi/3
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the part not need from this area, is the area of the triangle formed by the segment. This is an isosceles triange, since 2 of the legs are equal to the radius.
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The area of this triangle = 1/2*Base*height
if you draw a perpendicular from the central angle to the segment,dividing the central angle, this line would be defined as the height.
What you now have is two 90:60:30 triangles. The height=4*sin(30)=2
The base of one of the triangles is =4*cos(30)=4%2Asqrt%283%29%2F2=2*sqrt%283%29.
since there are two triangles forming the larger triangle, the base of the bigger triangle is 2*2*sqrt%283%29=4*sqrt%283%29
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So the area of this triangle=1/2*base*height= %281%2F2%29%2A%284%2Asqrt%283%29%29*(2)=4*sqrt%283%29
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So the desired area =
arc area - undesired triangle = 16*pi/3 - 4*sqrt%283%29
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you can do the math to get the actual value