Question

It is about finding area want help in part b?

Answer 1

Given triangle ABC is a equilateral triangle, one interior angle is `pi/3`.

Since the 3 arcs are equal,
`/_CAB=pi/3`
`2alpha = pi/3`

Since `/_AOB = 2pi/3` and `AO=OB=r`,
Triangle AOB is a equilateral triangle.
`:. r=4`

Substitute `2alpha = pi/3` and `r=4` into the answer for (a)(ii),
Area of segment AB
=`1/2r^2(pi/3-sin(pi/3))`
= `1/2(4)^2(pi/3-sqrt(3)/2)`
= `(8pi)/3-4sqrt(3)`

Area of triangle ABC
=`1/2a b sinC`
=`1/2(4)(4)(sin(pi/3))`
=`4sqrt(3)`

Area of shaded region
= Area of triangle ABC - 3 Area of segment AB
=`4sqrt(3)-3((8pi)/3-4sqrt(3))`
=`4sqrt(3)-8pi+12sqrt(3)`
=`16sqrt(3)-8pi`