regular hexagon abcdef is inscribed in a circle p with a radius of 12 centimeters.
calculate the exact areas of circle p and regular hexagon abcdef.
find the exact area of the shaded region shown in the image above.
imagine circle p with regular inscribed octagon abcdefgh, rather than regular hexagon abcdef. in two or more complete sentences, describe the effect the number of sides of the polygon would have on the area of the shaded region.

Step-by-step explanation:

A) Calculate the exact areas of circle P and regular hexagon ABCDEF.  

Solution:

Area of circle = πr²

where r= 12 cm

Area of circle = (3.14)(12)²

Area of circle =(3.14)(144)

Area of circle = 452.16 sq cm

Area of hexagon = 3√3 *(side)²/2

Since hexagon is consist of six identical equilateral triangles the side is same as the vertex to center. In this case it is same as the radius

Area of hexagon = 3√3 *(12)²/2

Area of hexagon = 3√3*(144)/2

Area of hexagon = 374.12 sq cm

B) Find the exact area of the shaded region:

Area of shaded region = Area of circle - Area of hexagon

Area of shaded region = 452.16 - 374.12

Area of shaded region = 78.27 sq cm

C) Imagine circle P with regular inscribed octagon ABCDEFGH, rather than regular hexagon ABCDEF. In two or more complete sentences, describe the effect the number of sides of the polygon would have on the area of the shaded region.

Since octagon has 8 sides so it will occupy more area than hexagon. The shaded area will then reduce. The polygon area increases, the shaded area decreases

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step-by-step explanation: