a regular hexagon is inscribed in a circle with a diameter of 8 inches. what is the perimeter of the hexagon? what is the area of the hexagon?
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From the middle of the hexagon to one of its edges will be 4.
A hexagon can be formed from 6 equilateral triangles
The area of one equilateral triangle in this case is:
A of hexagon = 8 * 6sin(60)
P = 4*6 = 24
given that its inscribed that means that it produces a hexagon that can be divided into 6 triangles. the triangles are all isosolese and the smaller part is actualy one face of the hexagon. if you drow out how this actuall looks you discover that the acute angle formed is 300/6 = 60 degrees. you can find the smaller side by simply using x/2 = 8sin30 , you use 30 degrees because you want a right trianlg ethe only way to do that is to cut it in half and produce a right triangle with acute angle 30 degrees and hypotonuse 8. Amazingly this side is 8 so the perimiter is 8*6 = 48. To find the area you can find the hight of one of the triangles which is h = 8cos30 ~~ 6.92.. then fidn the area of one triangle A = .5bh where base would be the x we found earlier so are of one triangle is A = 27.71. mustiply this by 6 to get the final answer of 166. This is however a shortcut of doing all of this useing two equations, i forgot the equations but most likely your teacher might know them or probably has already told you about them, in which case i would reccomend using the formula's since they are much faster and there is less chance of making a mistake.