Question

How do I find the area of the shaded region?

Answer 1

Part a) The area of the shaded region is
`A=25((2\pi)/(3)-√(3))\ cm^2`

Part b) The area of the shaded region is
`A=18(3\pi-2√(2))\ cm^2`

Explanation:

Part a) we know that

step 1

Find the area of the sector

we know that

The area of a circle subtends a central angle of 2π radians

so

using proportion

Find the area of the sector by a central angle of π/3 radians

`(\pi r^2)/(2\pi)= (x)/(\pi/3) \\\\x=(\pi r^2)/(6)`

we have

`r=10\ cm`

substitute

`x=(\pi (10)^2)/(6)\\\\x=(50\pi)/(3)\ cm^2`

step 2

Find the area of triangle

The area of triangle is equal to

`A=(1)/(2) (10^2)sin((\pi)/(3))`

Remember that

`(\pi)/(3)=60^o`

so

`A=50((√(3))/(2))=25√(3)\ cm^2`

step 3

Find the area of the shaded region

The area of the shaded region is equal to the area of the sector minus the area of isosceles triangle

so

`A=((50\pi)/(3)-25√(3))\ cm^2`

Simplify

`A=25((2\pi)/(3)-√(3))\ cm^2`

Part b) we know that

step 1

Find the area of the sector

we know that

The area of a circle subtends a central angle of 360 degrees

so

using proportion

Find the area of the sector by a central angle of 135 degrees

`(\pi r^2)/(360^o)= (x)/(135^o) \\\\x=0.375\pi r^2`

we have

`r=12\ cm`

substitute

`x=0.375\pi (12)^2\\\\x=54\pi\ cm^2`

step 2

Find the area of triangle

The area of triangle is equal to

`A=(1)/(2) (12^2)sin(135^o)`

`A=72((√(2))/(2))=36√(2)\ cm^2`

step 3

Find the area of the shaded region

The area of the shaded region is equal to the area of the sector minus the area of isosceles triangle

so

`A=(54\pi-36√(2))\ cm^2`

Simplify

`A=18(3\pi-2√(2))\ cm^2`