How do i find the area of the shaded region?

Part a) The area of the shaded region is

Part b) The area of the shaded region is

Step-by-step 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

we have

substitute

step 2

Find the area of triangle

The area of triangle is equal to

Remember that

so

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

Simplify

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

we have

substitute

step 2

Find the area of triangle

The area of triangle is equal to

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

Simplify