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Copy and complete the tables for the regular polygons shown. In these tables,$p$ represents the perimeter and $A$ represents the area.$\begin{array}{|c|c|c|c|}\hline r & {a} & {p} & {A} \\ \hline 6 & {?} & {?} & {?} \\ \hline\end{array}$

the required value of $a$ is ${3}$ $p$ is ${18 \sqrt{3}}$ and $A$ is $27 \sqrt{3}$

Geometry

Chapter 11

Areas of Plane Figures

Section 4

Areas of Regular Polygons

Polygons

Surface Area

Circles

Piedmont College

Cairn University

University of Michigan - Ann Arbor

Utica College

Lectures

12:03

In mathematics, a theorem …

07:43

In mathematics, the tangen…

Copy and complete the tabl…

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gives us a regular polygon of three sides. In this case, it's the triangle. So all sides are gonna be equal cause it's regular. And I've just added a couple more notations are labeled all three cornice. I've also labeled the Points X is the center of the circle. In our was the midpoint for May to bay so little X is gonna be the land of 80 R. And little lazy apathy. Um, to the apathy, um is the perpendicular distance from the origin of the circle to the midpoint of the one side of the public gun. So we're given that the radius of the Circle R is equal to six and we want to find the athm, the perimeter of the polygon on the area of the polygon. So first things first will find the athm. So what we can do is use trigonometry with news that the sign rule Stein Oh, oh, uh, we'll find Well, that's right. It was sign very every so we can rearrange Ferree. So a really cool It's all right. Okay. Hey, this ah times are so we'll need to find the angle. A. So what we can do is we know that the angle a next be it's guns equal 360 divided by three, an ankle under the 20 degrees. We know that because we scroll up and imagine So that's that's the angle XB. So if you want to draw and three sides triangles, you get three triangles in the center and you know the degrees in the circle of 3 60 So you divide it by how on the sides or if you will triangles, and that will give you the angle of each 120. So we know that the angle X or we can do no it as a, uh, an equal half of that. Is it the straight line down there to the right angle So unequal don't ever to ankles 60 degrees. Careful, we know that a so all angles in a triangle is gonna tow 1 80 degrees. So that means a is gonna equal 30 degrees, as we have 90 plus 60 could be 1 57 degrees remaining. So you pluck that into this equation, you should get a equals three. So that's that. And then now we need to find the perimeter. So perimeter is going to be the number of sides. So in this case is three sides come up for the triangle times the length of one side. That's going to be a big and we know a B is improvements to our So is the equivalent of eight R. Sorry, there's three phones to they are not to show you why that is the ghosts of the midpoint. Just the midpoint. It's going to be two of those to make hate today. That means is 68 NATO. We know it is little axe. And again you sign rule. So we rearrange the thanks in this little of equation here, so use this side of the equation and that sign. Thanks. Uh huh. That's gonna equal three or a three. When you put these numbers in, man for a he's gonna evil. 18 33 When you plug the numbers into this equation here for big and how old is left. As I said, the Arab other gun which will call it Hey, that's all the apathy sounds a perimeter we pluck in Lawrence's Hmm P uh, and a. So that's gonna be, uh, times three. I'm 18 3 27 through three

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