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Find the area of each polygon.A regular hexagon with perimeter 72

the required area $A$ of the regular hexagon with perimeter 72 is 216$\sqrt{3}$

Geometry

Chapter 11

Areas of Plane Figures

Section 4

Areas of Regular Polygons

Polygons

Surface Area

Circles

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right hex again. This inscribed inside of circle with radius are just on the diagram of what I would look like. There's a brooks kept, so the APA theme will be here. So that's gonna be the perpendicular distance from the origin to the center of the side. The polygon, they're X is the distance from P to the midpoint and little eyes, the radius paying here where they took society, the circle. And we need to find the perimeter. So given the perimeter was seventy two and we can immediately equate that because we knew the former That is gonna be the number of sites which is six times by the length of one sciences he keep. You can see that peak who is just two times that schools are is the midpoint that's gonna be sick. I was too an ankle twelve. Thanks That we can then work our angles to figure out the size of X. So the angle of okay, x Q Simply going to be the degrees in a circle three, then sixty divided by six Munchen with the drop A with these radius lines, I mean, a round circle will have six triangles like that Sure that So it's simply three and sixty. Divided by six is gonna get sixty degrees to the angle X, which is gonna be this angle here. That's gonna be exactly half of that. As you can see, it's split by the apathy. So in the people sixty degrees, three side, what about two? That's C degrees. So we can then use sign role to quit that. But before we do it, we can figure out the angle of peace. So that's not P for the perimeter. That's little pee there. So that's gonna we know that there's one hundred and eighty degrees in a triangle. So with one eighty minus the Angola, which is ninety minus the angle at X, which were just noticed thirty degrees, which will equal sixty degrees. And now it can you sign role so side of hey again, That's not the perimeter and equal sign o X X equals sign. Well, uh, we can rearrange these to find a and so it's gonna be a sign of hay plans by X. So sorry, that's gonna be big accent. Yeah, a little X, which we know those angles now that we've just figured out above here. That's gonna hungry. Found out. Um, what X is gonna equal? So that's going to be six three. Just her line. Do you that we know that axes six. Because if we rearrange this top equation head, we know that twelve x four seventy two so x sex. So we conclude this into our area, but the public grants That's really call that a b so off of the APP thing. So this is P for the permits and little pay. So what? There's in? Oh, I'm saying it's great. Already finds by anti to print and people two hundred sixteen breakthrough That's final inside.

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