Question

All shapes in the picture are half-circles. If the radius of the smaller half-circle is 1, what is the area of the colored region?

The tic marks represent the center of the smaller half-circles.

The center tic mark is also the center of the larger half-circle.

Answer 1

`((7pi)/6-sqrt3/2)~~2.799 " unit"^2`

Explanation:

let `r` be the radius of the smaller semi-circles and `R` be the radius of the larger semi-circle,
given `r=1, => R=2`
Area of the larger semi-circle `=A_1=(pi*2^2)/2=2pi`
Area of each smaller semi-circle `=A_2=(pi*1^2)/2=pi/2`
As `OA=OB=AB=1, => DeltaOAB` is equilateral
Area of each equilateral triangle `=A_3=sqrt3/4*1=sqrt3/4`
Area of each segment `(OB) =A_4=pir^2*(60)/(360)-A_3`
`=pi/6-sqrt3/4=(2pi-3sqrt3)/12`
Now,
Green area `=A_1-3A_2+2A_3+4A_4`
`=2pi-(3pi)/2+(2sqrt3)/4+(4(2pi-3sqrt3))/12`
`=2pi-(3pi)/2+(2pi)/3+sqrt3/2-sqrt3`
`=(12pi-9pi+4pi)/6-sqrt3/2`
`=(7pi)/6-sqrt3/2`
`~~2.799 " units"^2`