Question

What is the area of the shaded region? What is the answer to the nearest tenth of a square unit?

Answer 1

`34.75`square units

Explanation:

Where `theta=360/"number of sides"=360/6=60^@`

Radius of Circle
`r=sqrt(s^2/(2-2costheta))`
`r=sqrt(8^2/(2-2cos60^@))`
`r=sqrt(64/(2-2(1/2)))=sqrt(64)=8`

Thus the radius is also 8.

Area of Hexagon:
`A=("number of sides")((r^2sintheta)/2)`
`A=(6)((8^2sin60^@)/2)=192sqrt(3)`

Area of Circle:
`A=pir^2=64pi`

Area of Shaded Region:
`("Area of Circle")-("Area of Hexagon")=64pi-96sqrt(3)/2=34.75`square units