Question

The regular polygon has radius 9 m. Find each angle measure to the nearest tenth of a degree, each linear measure to the nearest tenth of a meter, and the square measure to the nearest square meter.

Answer 1

AOX
AOB
OX
AB
the peremiter
the area

Answer 2

I'm assuming that's a poorly dragon octagon.
All my work after this point will be based on that assumption.

Draw a dot in the center of the figure. Now, draw a line from that dot into each corner. Now, draw a circle around the dot. As you can see, the circle is divided into 8 equal parts, 8 equal triangles. So this tells us that each triangle will have 360/8 = 45 degrees on 1 corner. I tried to draw it out, but I suck a drawing.

So what happened is that I split the octagon into 8 different isosceles triangles, with each one having an angle of 45 degrees and an angle of (180 - 45)/2 = 67.5 degrees each. Each corner has 2 of these of these angels, so each corner has an angle of 67.5 * 2 = 130 degrees (I'm assuming that is what's meant by finding the angle, because that's the only angle I see). I'm hitting enter now and continuing in a different post because waiting 5-10 minutes can be boring. Keep waiting.

Answer 3

I don't know what's asked by a lot of the other stuff, so I'll be hitting the perimeter, the area, and the angle of each corner (which I just did).

Onto the perimeter/area:
Okay, so we've split the octagon into 8 different triangles. We know the following about the triangles:
1. They're isosceles
2. They have angles of 45, 67.5, 67.5 degrees
3. They have 2 sides of 9 m.

Since it's a triangle, I can't mess up the drawing, so I'll post a crudely made picture onto imgur.
http://imgur.com/INXbmQZ
The sides pointed by the red arrows are the ones with the length of 9 m.
The red angles are the ones of 67.5 degrees.
The blue angle is the angel of 45 degrees.

Now where do we go from here?
Let's find the length of the unknown side (side with no arrow). Why? This side is the one that's part of the octagon, and I know that the process of finding the length of this side will allow me to solve for the area.

So how do we solve for the unknown side? The obvious answer that jumps out first is Pythagorean Thrm. But we don't have a right triangle. But what we do know is that we can split an isosceles triangle into 2 identical right triangles (you can probably google a drawing of this but it's kind of obvious). And we do this by starting from the corner of the 45 degree angle, and drawing a line straight down the middle. Another imgur link to show this:
http://imgur.com/8gRlSFE
(Drawing is horrible)

I depicted this split by drawing a yellow line. The 2 grey squares are to show you that the line forms 2 angles of 90 degrees when they meet (a line is 180 degrees. Drawing a line straight down the middle is halving it. 180 degrees => 2x 90 degree parts). So we have a right triangle now with the angles: 67.5, 90, 22.5.

Where did this 22.5 come from? Since we formed the triangle by splitting it into half, the triangle will contain half of the angle we split it in half from, so 45 => 22.5.

Doing basic math, we can confirm it is a valid triangle with 180 degrees.

So now we have a triangle with 3 known angles, 1 known side. We can do basic trig to determine the other 2 sides (need both unknown sides for area!). We can see that 9 m is the hypotenuse. Let's solve for the bottom side 1st. The bottom side is adjacent to the angle that's 67.5 degrees. We have an angle, and we have a hypotenuse length. Which means that we can solve for it with basic trig (sin, cos, tan). Note how I said adjacent, so we're going to be using cos. Set up the cos equation like so:
cos(67.5) = x/9
where x is the unknown side.
Multiply both sides by 9, and we get: roughly 3.444 m. 16 of these sides make up the octagon (remember, we split the octagon into 8 equal triangles, and then halved it, leaving 16 equal parts). Multiplying by 16, we get the perimeter of the octagon, which is ~55.1064 m.

We have 2 sides of the triangle. We can solve for the third using basic trig or Pythag. Since I feel that Pythag is easier, let's do that.

We have the length of the hypotenuse and the length of 1 side. We can solve for the square of the other side by subtracting the side's length squared from the square of the hypotenuse's length. We get: 69.138 as the square of the unknown side's length. But this is the square, so we take the square root of that number to get: 8.135 m. As we can see from the 2nd imgur link I posted, this is our triangle "height". We now have all 3 sides of the triangle.

As I've stated before, the right triangle we've been working on is 1/16th of the actual octagon. So, if we find the area, and multiply it by 16, we get the total area of the octagon. The area of the triangle is (0.5)(8.315)(3.444). When we multiply that number by 16, we get roughly 229 m^2 as the area.

tl;dr:
perimeter: 55.1064 m
area: 229 m^2 (there's some rounding error)

Answer 4

Definitely tl;dr. But this man just spent all that time writing all of that out. He deserves a medal. No, literally. A medal. o: good for you, sir. I am fanning you. Nobody takes that much time and effort for a single answer.