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Home / Assignment Help / Find the area of the shaded region. Round answers to the nearest tenth. Assume all inscribed polygons are regular.

# Find the area of the shaded region. Round answers to the nearest tenth. Assume all inscribed polygons are regular.

We can calculate for the area of the shaded region by 1)
calculating the area of circle, 2) calculating for area of triangle, and 3)
calculating for area of the segment below the triangle. Then subtract 2 and 3
from 1.

1) Area of circle, Ac = π r^2

Ac = π (6)^2

Ac = 113.10

2) Area of triangle, At

The triangle is an equilateral triangle with angles on each
corner equal to 60 degrees. Meanwhile, the 3 angles at the center is 120
degrees each since a circle is 360 degree. We know that the radius (line from
centerpoint to corner) is equivalent to 6. Using the cosine law, we can calculate
for the length of one side.

s^2 = 6^ + 6^2 – 2 (6) (6) cos 120

s^2 = 108

s = 10.4

Since this is an equilateral triangle therefore all sides
are equal. The area for this is:

At = (sqrt3 / 4) * s^2

At = 46.77

3) Area of segment, As is given as:

As = [ (r^2) / 2 ] ( θ rad – sin θ )

As = [(6^2) / 2] (120 * π / 180 – sin 120)

As = 22.11

Therefore the area of shaded region is:

A = 113.10 – 46.77 – 22.11

A = 44.22 