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.
Answers
Given : Regular hexagon ABCDEF is inscribed in a circle P with a radius of 12 centimeters.
To find : area of the shaded region
Solution:
in Hexagon if we join two vertex of a side with center then we get equilateral triangle
Hence radius = side of hexagon = 12 cm
Area of Hexagon = 6 * (√3 / 4) 12² = 216√3 cm²
Area of circle = π* 12² = 144π cm²
Area of shaded region = 144π - 216√3 cm² = 78.3 cm²
octagon => center angle = 45°
area of each triangle = (1/2) 12 * 12 Sin45° = (1/2) 12² (1/√2)
area of octagon = 8 * (1/2) 12² (1/√2) = 407.23 cm²
area of circle = 144π cm² = 452.39 cm²
area of shaded region = 45.16 cm²
area of polygon with n sides = n *(1/2) * r² sin(360/n)
Area of shaded region in octagon would be less than in case of hexagon.
if number of sides of polygon would increase in a given circle then area of polygon will increase hence area of shaded region will decrease.
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