Question

Regular hexagon ABCDEF is inscribed in a circle P with a radius of 12 centimeters.

Answer 1

a. Calculate the exact areas of circle P and regular hexagon ABCDEF.
b. Find the exact area of the shaded region shown in the image above.
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.

Picture: http://i.gyazo.com/d5c9ca7313eeab87b4c79e300183b0bc.png

Answer 2

All I know so far is that 360/6 = 60, so \[\theta = 60°\]

Answer 3

a. area of the circle = pi r^2 where radius r = 12
the area of the hexagon is 6 times the area of the triangle shown in the daigram

Answer 4

there are 6 of these triangles in the hexagon
area of one triangle = 0.5 * 6 * 6 sqrt3

Answer 5

the area of the shaded region = area of circle - area of the hexagon

Answer 6

they want exact answers meaning they want the answer in terms of square roots and pi

Answer 7

So the area of the circle would be π12^2 equaling out to 144π. When you say the area of the hexagon is 6 * the area of the circle, would that mean that it would be 864π?

Answer 8

i made one mistake - the area of the triangle is 6 * 3 sqrt 3 = 36sqrt3

Answer 9

area of the circle is pi * 12^2 = 144pi
and the area of the hexagon = 6 * 36 sqrt3 = 216sqrt3

Answer 10

so shaded area = 144pi - 216sqrt3

Answer 11

you can work this out on your calculator if you like and give the answer correct to 2 places of decimals

Answer 12

do you follow that ok?

Answer 13

Wow you're a lifesaver. Thank you so much man :)
And yeah I follow it alright. I have a much better understanding on this now.

Answer 14

good
for c an octagon would have a greater area than a hexagon and so would fill more of the circle. Therefore the shaded area would have a smaller area than with the hexagon.