Apothem: find area, perim. of reg. octagon w/ apothem of 1

[MsDeb52]

I have a discussion question, that I have actively researched, but I think that the question is missing a component. I am not yet comfortable enough with my knowledge to be the first one to point this out. This is the question:

What is the area and perimeter of a regular octagon with apothem of 1? With radius of 1?

Don't I need more information. I have an apothem of 1, or a radius of 1, and I know that it is an octogon (8 sides). But, I lack information to complete the formula for area: A = 1/2 a p.

Am I overlooking something simple? I am using Prentice Hall Mathematics - Geometry, Chapter 7.

Thank you very much for taking the time to look at this.

 

[galactus]

Re: Apothem

There is enough info.

Knowing the apothem, the area can be found by using the formula:

\(\displaystyle A=a^{2}\cdot n\cdot tan(\frac{\pi}{n})\)...in radians.

Once the area is found, use the formula \(\displaystyle P=\frac{2\cdot\text{area}}{a}\) to find the perimeter.

a=length of apothem and n=number of sides.

 

[MsDeb52]

First, thank you for your response. It was not understood, but it was appreciated.

Second, your statement at the bottom was intended as a joke, correct? "Your mom...."

Third, I am 57 and my last encounter with trig was in 1994, when I took a class. I don't remember anything about tangents, etc., and my Geometry Book does not have any trig components. And, as we are retired and travelling in our RV, and all of my possessions are in storage, I don't have access to my trig book.

Fourth, is there any other possible way, based on the level of instructino indicated by my textbook. Seriously, the section has three pages, introduces us to the concept of apothem, and the book features no examples, or problems, similar to the discussion question, and all the examples are based on the "special triangles - 30,60,90, or 45,45,90".

This is my conundrum. The book is on one level and the discussion question appears to be on a greatly advanced level.

Anyway, your assistance, and kindness was greatly appreciated.

Deborah

 

[Denis]

What is the area and perimeter of a regular octagon with apothem of 1? With radius of 1?
Don't I need more information. I have an apothem of 1, or a radius of 1, and I know that it is an octogon (8 sides).

That's 2 different questions:
What is the area and perimeter of a regular octagon with apothem = 1 ?
What is the area and perimeter of a regular octagon with radius = 1 ?
The apothem is shorter than the radius; go here for clarification:
http://mathworld.wolfram.com/Apothem.html

For you to "see" what going on:
- draw an octagon
- draw 8 lines: from the center of the octagon to the 8 corners (or vertices)
- you'll then get 8 identical triangles
The area of the octagon is 8 times the area of 1 of those triangles; see that?

Use google to get sites that'll show you examples; you'll find it's actually quite easy...

 

[MsDeb52]

Thank you for taking the time to look over my problem.

I did try the Google help sites first, including Dr.Math (from Drexel University). By the way, the link you gave me didn't work. Oh, well....

Anyway, I can comprehend that I just need the area of 1 triangle, then multiply by 8

However, the only formula I know is A = 1/2 base times height

If the only information I am given is that it is an octagon, and I have either an apothem of 1, or the other part of the same problem, a radius of 1, I still have insufficient information to get an answer. Each of the 8 triangles are 45 degrees at the center, and the base angles are 67.5 degrees each. So, I can't use the "special triangles" or anything I can think of to obtain the necessary other component for the formula.

Therein lies my problem - there is no numerical number assigned to the base of the triangle, the side of the octagon. So, I can't divide it in half, and then plug into my formula.

So, thank you for taking the time to consider this issue. I appreciate it. I hope all is well with you, and I appreciate your help. It means a lot.

Again, thank you.
Sincerely,
Deborah Clement

 

[Denis]

OK MsDeb; lets take one of the 8 triangles inside the octagon;
call the octagon's center C; label the side (on octagon) AB.
So triangle ABC has angle of 45 degrees at C (360/8), right?
And the height (apothem) of the triangle = 1 ; that's given.

Drawing the apothem line makes 2 right triangles, right?
So we can use one of those right triangles: angles 22.5, 67.5 and 90, plus side opposite 67.5 = 1 (given).
The other leg (non-hypotenuse) of that triangle = apothem * TAN(22.5) = 1 * TAN(22.5) = .4142 (rounded).

Since this is half the octagon's side length, the side length = .8284 (rounded).

So you can now easily get the Area and Perimeter.

Hope the Easter Bunny leaves you 2 Eggs :wink:

 

[TchrWill]

I have a discussion question, that I have actively researched, but I think that the question is missing a component. I am not yet comfortable enough with my knowledge to be the first one to point this out. This is the question:

What is the area and perimeter of a regular octagon with apothem of 1? With radius of 1?

Don't I need more information. I have an apothem of 1, or a radius of 1, and I know that it is an octogon (8 sides). But, I lack information to complete the formula for area: A = 1/2 a p.

Am I overlooking something simple? I am using Prentice Hall Mathematics - Geometry, Chapter 7.

Thank you very much for taking the time to look at this.


POLYGONS

A polygon is a plane figure with three or more line segments and angles that are joined end to end so as to completely enclose an area without any of the line segments intersecting.

A convex polygon is one where the line segments joining any two points of the polygon remain totally inside the polygon, each interior angle being less than 180º.

A concave polygon is one where one or more line segments joining any two points of the polygon are outside of the polygon and one or more of the interior angles is greater than 180º. The inward pointing angle of a concave polygon is referred to as a reentrant angle. The angles less than 180º are called salient angles.

A regular polygon is one where all the sides have the same length and all the interior angles are equal.

A diagonal is a straight line connecting any two opposite vertices of the polygon.

Polygons are classified by the number of sides they have.

No. of sides.........Polygon Name
......3.....................Triangle
......4..................Quadrilateral
......5....................Pentogon
......6....................Hexagon
......7....................Heptagon
.....8......................Octagon
.....9......................Nonagon
....10.....................Decagon
....11....................Undecagon
....12....................Dodecagon
....13....................Tridecagon
....14....................Tetradecagon
....15....................Pentadecagon
......n........................n-gon

Regular Polygon Terminology

n = the number of sides

v = angle subtended at the center by one side = 360/n

s = the length of one side = R[2sin(v/2)] = r[2tan(v/2)]

R = the radius of the circumscribed circle = s[csc(v/2)]/2 = r[sec(v/2)]

r = the radius of the inscribed circle = R[cos(v/2)] = s[cot(v/2)]/2

a = apothem = the perpendicular distance from the center to a side (the radius of the inscribed circle)

p = the perimeter = ns

Area = s^2[ncot(v/2)]/4 = R^2[nsin(v/2)]/2 = r^2[ntan(v/2)]

The formula for the area of a regular polygon is also A = (1/2 )ap = (1/2)ans, where a is the apothem, p is the perimeter, s is the side length and n is the number of sides..

The sum of all the interior angles in a polygon is 180(n - 2)

The sum of the exterior angles in a polygon is 360º.

The internal angle between two adjacent sides of a regular polygon is given by 180(n - 2)/n

The external angle between any side and the extended adjacent side of a regular polygon is given by 360/n.

You might be interested in why the sum of all the interior angles of a polygon is 180(n - 2).
Consider first the square, rectangle and trapazoid. Draw one ofthe diagonals in each of these figures.
What is created is two triangles within each figure.
The sum of the interior angles of any triangle is 180 deg.
Therefore, the sum of the interior angles of each of these 4 sided figues is 360 Deg.
Now consider a pentagon with 5 sides that can be divided up into 3 triangles.
Therefore, the sum of the interior angles of a pentagon is 540 Deg.
What about a hexagon. I tink you will soonsee that the sum of the interior angles is 720 Deg.
Do you notice anything?
n = number of sides........3........4........5........6
Sum of Int. Angles.........180....360....540....720
The sum of the interior angles is representable by 180(n - 2).

Consider also the sum of the exterior angles.
Each exterior angle is 180 - 180(n - 2)/n = (180 - 180n + 360)/n = 360/n.
Therefore, the sum of the exterior angles is 360n/n or 360 Deg.