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Three regular polygons are inscribed in circles with radii $1 .$ Find the apothem, the perimeter, and the area of each polygon. Use $\sqrt{3} \approx 1.732$ and $\sqrt{2} \approx 1.414$ .

the required value of the apothem is 0.707the required perimeter is $5.656$the required value of the area $A$ of the pentagon is 2

Geometry

Chapter 11

Areas of Plane Figures

Section 4

Areas of Regular Polygons

Polygons

Surface Area

Circles

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we will start out by drawing a square and inside the square we have a inscribed right triangle. There we go and things to remember within a square you have 90 degrees all the way around. I'm in a redraw, our red right triangle, and when the right triangle is inscribed it by sex the 90 degrees. So we now have a 45 45 90 triangle, and the standard 45 45 90 triangle is both legs our X because they are congruent. And then the high pot news or the longest side, is X radical, too. Now, in the original problem, the length of the inscribed high pot news is one. So I set the standard, which is right here, X radical to the given for a high pot news. I always say standard equals given, and they give you the value of one. So you saw that for acts by dividing both sides by radical too. And when you simplify the radical twos, cancel. So X is equal to one over radical too. But you cannot have a radical in the denominator, so we multiply by radical to over radical too, and that simplifies in the numerator Too radical, too over radical. Four The denominators A perfect square. So numerator stays radical to over square root. A four is too and the decimal value of radical to over to is 0.707 if you want the dustman. So I go back to my original drawing and radical to over two is half of the original square. So then I multiply that by 2/1 to get the other half of the square. The twos cancel so the side length of the square is radical too. I'm gonna go and label that and each side is radical too on the square. And then one of the questions they asked for was area of the square. So besides squared or radical, too times radical too, which is radical for which then is to for the area. So we have the possum, we have the area. Then final thing we needed was perimeter. Every side of the square is radical too. And there's four sides, so we can add it or we can multiply. Since there's four of them and four sides for radical too. If you want the decimal value of that radical two times four is approximately 5.656 So you have the perimeter, the air

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