Question

Find the area of the shaded portion. Show all work for full credit.

Answer 1

Answer:

Area of the shaded region = 24.67 square units

Step-by-step explanation:

To find the area of the shaded region. First, we need to find the area of the sector.

We have formula to find the area of the sector.

Area of sector of a circle = $\frac{central angle}{360} *\pi r^2$

From the given figure, radius (r) = 6 and central angle = 120

Now plug in these values in the above formula.

=$\frac{120}{360} *\pi * 6^2$

= $\frac{1}{3} *\pi *36 = 12\pi$

The area of the sector of the circle = 12π

Now we have to draw line bisector from the external point and that divide into two right triangle in ratio of angle 30 -60 -90

Now we use the ratio of 30-60-90 degree and find the base of the triangle.

The ratio of sides are 1x:√3x:2x

Given: 1x = 6

So the base is √3x = 6√3

Now we find the area of 1 right triangle = $\frac{1}{2} *base * height\\= \frac{1}{2} *6\sqrt{3} *6\\= 18\sqrt{3}$

So the area of two triangles = 2*18√3

= 36√3

To find the shaded region we have to subtract the area of the sector form the area of the two triangles

= 36√3 - 12π

The value of π = 3.14

= 36√3 - 12*3.14

= 62.35 - 37.68

Area of the shaded region = 24.67 square units

Answer 2

Given the figure:

The circle has a radius of 6 units. An included angle of 120 degrees. The shaded portion is outside the circle with sides tangent to the circle. 

We can draw two right triangles by drawing a line bisecting the included angle. Then, we use the Pythagorean theorem to solve for the measurement of the two sides. 

let x = measure of the side of the shaded portion

tan (60) = opposite / adjacent
tan (60) = x / 6

solve for x, x = 10.39

let s = arc

s = rθ
s = 6 * 120*pi/180
s = 4*pi

then solve for the area of the shaded portion. <span />