# ABC is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of shaded reigon. Leave the answer in pie or surds form

2
by kudumba

2014-07-28T16:52:05+05:30

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Which area is shaded ?  is it outside the ABC and the inside the circle?
Equilateral triangle : side = 4 cm.  Each angle = 60 degrees.
Altitude = height of triangle =  AB * sin 60  = 4 *√3/2 = 2 √3 cm
Area triangle ABC = 1/2 * 4 * 2√3 = 4 √3  cm²

Area of circle = π r²  = π  (4/√3)² = 16π /3  cm²      as radius of circum circle = AB / √3

Shaded area = 16π/3 -  4√3   cm²

DO you need to derive the radius of circumcircle of ABC?
Let center of circle be O.  Md point of AB = D.  AO bisects angle A.  => in triangle OAB, angle OAB = 30 deg.  Draw a perpendicular from O onto AB intersecting AB at D.

Cos  angle A/2    =  AD  /  AO    =  AB/2  / r  = AB / 2r
cos 30 = √3 / 2  =  AB / 2r    =>  r = AB / √3

2014-07-28T18:24:04+05:30
Equilateral triangle : side = 4 cm.  Each angle = 60 degrees.
Altitude = height of triangle =  AB * sin 60  = 4 *√3/2 = 2 √3 cm
Area triangle ABC = 1/2 * 4 * 2√3 = 4 √3  cm²

Area of circle = π r²  = π  (4/√3)² = 16π /3  cm²      as radius of circum circle = AB / √3

Shaded area = 16π/3 -  4√3   cm²

DO you need to derive the radius of circumcircle of ABC?
Let center of circle be O.  Md point of AB = D.  AO bisects angle A.  => in triangle OAB, angle OAB = 30 deg.  Draw a perpendicular from O onto AB intersecting AB at D.

Cos  angle A/2    =  AD  /  AO    =  AB/2  / r  = AB / 2r
cos 30 = √3 / 2  =  AB / 2r    =>  r = AB / √3