Copy and complete the tables for the regular polygons shown. In these tables,
$p$ represents the perimeter and $A$ represents the area.
$\begin{array}{|c|c|c|c|}\hline r & {a} & {p} & {A} \\ \hline 4 & {?} & {?} & {?} \\ \hline\end{array}$

Answer

the required value of $a$ is 2$\sqrt{3}$ $p$ is 24 and $A$ is 24$\sqrt{3}$

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Geometry

Chapter 11

Areas of Plane Figures

Section 4

Areas of Regular Polygons

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Video Transcript

regular polygon that's inscribed inside a circle. I've just had this, um, actually points so see, on a heart to points whether me the circle andi b is the origin of the circle, excess is the angle and little x of the length of our to A So we need to find a which is known as the apathy. Um, so that's the distance from the origin to perpendicular to the middle of the side of the polygon on. We need to find the perimeter of the public gun in the air of the bulletin. So we're given the radius equals four. So what we can do is you first find the angles. So the angle see, be a show with this red line hair. So that's not angle here to that. So that's gonna be 360 degrees. Find it by six, which is gonna ankle. Sorry, may not need toe, so it's gonna be cool 60 degrees and reason. We know that because if you imagine during the rest of the radius lines the points of the circles, you imagine it's going to make six of those triangles on the degrees and circle of its A 3 60 So it's simply divide that by six together angle to another angle. X Gonna be exactly half of that because it splits it down the middle. With that right angle, there is 68 to equal 30 degrees. Therefore, we can find a because all the angles in the triangle at upto 1 88 People's 1 80 My goodness. 1960 home. Sorry. Lost 13 equal. 60 grace you go. So no become work on finding the apathy on what we can use for that is signed Roll. So we know that sign Havre equals sign. Okay. And right. Local sign X, it ranks. So what will you do and just focus on part of the equations that we know? So in this case, we can focus on their heart here to find a say, Hey, we re arrange that. And ankles. Okay, a list sign. Thanks. Times by X slightly. Oh, sorry. Yeah. So we used the are, actually. So that's the one we know. Sorry. Made him stay there. So it's times why are Because we know those angles? Because we don't know the the we don't know about your backs. So once we plug articles for And we know that our his 90 degrees we should get a going to be equals two, right? Three. So we know that perimeter what we need to find next. So that's gonna be the number of size, which is six times lent one side just going to be a c. We knew that a c Samos singing too eight. Ah, or ex. So thank you for coming to X equals 12 ax. Andi, we confined axe use in this sign rule from above. So thanks, people. Sign. Thanks. They were signing. Listen, uh, so that when you put those and get to Hey, you put that in. He's going to Eagle 20 full. Yeah, and then because we know A and P something's theory of the program was hopefully oh, times a. Okay. When you put those in, then you should get 20 full root

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