<PRE><B>Question 333857</B>
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*4^2}}}=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*sqrt(3)/2}}}=2*{{{sqrt(3)}}}.
since there are two triangles forming the larger triangle, the base of the bigger triangle is 2*2*{{{sqrt(3)}}}=4*{{{sqrt(3)}}}
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So the area of this triangle=1/2*base*height= {{{(1/2)*(4*sqrt(3))}}}*(2)=4*{{{sqrt(3)}}}
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So the desired area = 
arc area - undesired triangle = 16*{{{pi}}}/3 - 4*{{{sqrt(3)}}}

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you can do the math to get the actual value</PRE>