Talk:Apothems and Area

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Chris 15:03, 18 July 2012 (EDT)

I changed "midpoint of the side" to "midpoint of one side." Sorry I didn't catch this before, but one problem with the intro is that a circle is not a polygon. I would either remove the circle from the image or add a sentence relating the apothem to the radius of a circle that circumscribes the polygon.

The text next to Figure 3 needs better spacing. For example, add a space between "a=xcos30" and "Thus the area..."

Centroid: The centroid of a triangle is the intersection of the medians of the triangle. I would put a mouseover for median, describing it as a line connecting a vertex with the midpoint of the line opposite from it. The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint. Since a centroid is a point, it cannot be equal to the apothem, which is a line. The median of an equilateral triangle is congruent to the apothem.

Next to and in Figure 12. I would simply write 3r(1), not 3*r(1).

Jorin 09:28, 17 July 2012 (EDT) Maybe in the basic description, label the images with like Fig. 1 or Fig. 2 or something like that, and then when you are explaining how to find the area of the polygon, tell the reader which image is the more helpful. That would intertwine the images better with the text.

Maybe some more concrete steps in the first hexagon example? I don't know.

When deriving the formula for the angles of a regular polygon, just describe where the n - 2 comes from. Show how the example corresponds to the general formula.

In the "Wire Problem" section, paragraph 1, Maybe have "Students are often asked to solve the wired problem for the following scenarios" or something like that.

Change n to q in that section of the wire problem that we talked about.

I'm totally lost in the math with the p-gon and q-gon. I think if you define alpha, that would be really helpful. Make sure you define every variable and say why each equation is true explicitly.

Chris 09:14, 2 July 2012 (EDT) Noting the corrections you've already made, I have a few more suggestions.

Basic Description

  • P3S3 Remove this; finding the perimeter of the rectangle is superfluous and if anything only serves to confuse the reader.
  • Below the graphic: Remove bolded sentence "What are the dimensions of the rectangle?" and "Perimeter of rectangle: 12x + 2a" Combine the base of the rectangle and area of rectangle sentences.
  • Remove "What are the dimensions of the rectangle?"
  • You asked about consistency in representing math. I would stay away from the X to represent multiplication. I would change cos(30) to cos 30˚.

General Formula for Finding Area

  • P1S2: I realize now that my recommendation to change this sentence to "Since the figure is a regular polygon, the measure of one of its angles is..." causes for confusion with the graphic, because the angle we're referring to is actually MBC, which is only one of many angles shown in the diagram. I would delete this sentence above the diagram because you repeat it later after explaining the diagram.
  • When you solve for tan∠ABD, do you intend to have the math on the left and the commentary on the right? You include things like "Look to the image at left." The challenge here is the balance between being true to Emma's voice and making things clearer to the reader. Her check of the formula again uses a hexagon, as in the example above. It might make sense to use a different example of regular polygon in one of the two examples; I'm thinking that the first example might make the most sense.
  • I'm copying two suggestions I made earlier from the "Using the Apothem to Solve the Wire Problem" section since they have not been changed.
    • P1S2: To "so that the sum of the enclosed areas is minimized", add "or maximized." Isn't that the more common goal?
    • P3: You can't read the whole paper by clicking there; you get only the first page.
  • I noticed that you did change the apothem distance to 1/3 the height. I think you need to explain why it is 1/3, which relates to the centroid. See my earlier comments on that section.

Hey, Team! A few concerns..

  1. Do you see the page as high-school friendly? If not, what things could be added/better clarified to improve it?
  2. What math should be hidden?
  3. I am not very consistent with how I represent "math" on the page. Sometimes I use  \times , sometimes I don't, the spacing is kinda a mess, etc. What do you think looks best?
  4. Right now I only have one real application...any other ideas?
  5. please please please correct any spelling/grammar problems you see

just looking for any general advice/comments, this is a work in progress!

Chris 11:05, 28 June 2012 (EDT) Hey Alexis: Thanks for your work with Emma's page. I like both the subsections and the Wire problem addition. Is your idea of solving it using apothems your own? If so, that's even better!

Some of my suggestions below apply to doing more with Emma's original text. Some apply to yours.

Apothems and Area

Per our discussion yesterday, we want the basic description to include the beginning part of what you have now in the "More Mathmatical Explanation" up to the part on trigonometry.

What is an Apothem?

  • Please define apothem.

Basic Description

  • "as shown here." I assume you mean that the apothem is shown here, however, the last noun is "area", which is not shown here. The reference needs more clarity.
  • The text in the paragraph beginning with "If the regular polygon" and the next section's first paragraph (beginning with "Let's start with a simple..." can be integrated and combined with the graphics to make a stronger explanation of how the formula is derived.
  • Paragraph 2, Sentence 1 (P2, S1): "the triangles can be arranged to form half of a rectangle." I would change "arranged" to "unrolled"
  • I would end this section with the derivation of the formula, meaning 1/2ap, with p representing the perimeter of the polygon.

A More Mathematical Explanation

  • I would begin this section about where you have written: "But what is a?"
  • Please explain how you knew that the angle was 30˚ for the triangles used in the regular hexagon. (You might talk about all the triangles being equilateral and thus equiangular.) I might then use the properties of a 30-60-90˚ triangle to calculate the side from there.
  • "Let's check this formula by assuming x=1." Not sure what you are doing with this.

General Formula for Finding Area

  • P1S2: Change to "Since the figure is a regular polygon, the measure of one of its angles is..."
  • P2S2: I'd remove the sentence "Let's say that angle MBC..."
  • P2S3: Change to "We've connected center A to vertices B and C to create a triangle. We then dropped an altitude, creating line segment AD whose length is the apothem, a."
  • P2S8: Remove "And" from "ABC is half of MBC" and add a period. Then the Show WHy This Is True" can stand alone.
  • P4S4: "The tangent of angle ABD...ofer the adjacent side." Change "ofer" to "over."

Using the Apothem to Solve the Wire Problem

  • P1S2: To "so that the sum of the enclosed areas is minimized", add "or maximized." Isn't that the more common goal?
  • P3: You can't read the whole paper by clicking there; you get only the first page.

Eliminating Calculus

Top section looks really good.

  • P4S3: The apothem of the triangle is not equal to half the height. It is 1/3 the height, since the center point of an equilateral triangle is also the centroid, which is located 2/3 of the distance from the vertex to the midpoint of the opposite side. You can see from the diagram of the inscribed circle that r1 can't possibly be 1/2 the height of the triangle.
  • Make the appropriate adjustments in the ensuing math.

Hey, Emma! My name is Alexis and I am a rising sophomore at Swarthmore College. I really like your page, and I have some ideas for you! However, I know it is summer break. If you don't want to work on your page over the summer that is totally okay!


  • This is a great intro section! However, I'm not sure if you want to have an image here in text. The intro section is usually used to talk about the main image, and too many images too close together can confuse the reader. Instead, you might want to write out a definition for an apothem.

Basic Description

  • The image from above might be more helpful if moved to this section!

More Mathematical Explanation

  • This section is full of great information and math! The reader might be overwhelmed by such a big paragraph. It might be better to separate the MME section into smaller sections with subheadings. It is also helpful to take the math formulas "out of text" (putting the formulas on their own line). You could also add some more images of the steps that you are explaining to the reader.

Check out this page's MME! They did a great job creating subsections:

More Ideas

  • Lastly, I was considering adding a "Why It's Interesting" section with some real world applications of apothems. Since you did such a great job explaining the formula behind apothems, we should show the reader where else they can apply the formula!

Let me know what you think of my ideas! Feel free to e-mail me at or just write on my user page! If you want, we can work on this page together. If not, I would love permission to add some of these ideas! Alexis Z.

I'm experiencing some technical difficulties... In my very first paragraph I'm trying to put an image in a balloon. When I mouse over the balloon it just says it can't locate the file. I've uploaded it twice and it's definitely there. Help me (please). Emma F. 13:18, 17 April 2012 (EDT)

I got the mouse-over to work using the syntax, {{EasyBalloon|Link=here|Balloon=[[Image:Hextorec.png]] A visual description of the apothem's role in area.}}. I didn't change it on your page, because I wasn't sure what you actually wanted the balloon to look like, so try that and see if it's what you want. However, I don't really think that section belongs in a mouse-over, as you'll see:
I love your page! You explain your ideas and the math in a way that's clear, friendly, and interesting. This was really fun to read. I only have a few points where I'd suggest making changes. I know this looks like a long list, but it's mostly explanations of formatting points:
  • A general point first: Try to be consistent in how you choose to write variables or mathematical language. You often switch between calling the same variable "x" and "x". While these are technically the same letter, they look very different, and this can be confusing. What I tend to do is refer to a variable in-text with italics, x, and revert to the math-x only in a complicated equation that requires math script: \int_0^\pi\frac{3-t}{x}. done
  • The Balloons in your intro (on "here") and in your more mathematical explanation ("ABC is half of MBC") would work better as either hidden sections or just regular parts of the page. Balloons are most effective as brief asides, definitions, or clarifications, like the one you use in your basic description to define "apothem."
    • For the balloon in your intro, I suggest putting that image directly into the body of the page, perhaps scaled-down to make the page look less crowded, and just ending that opening blurb, "This can be explained visually: [[your picture]]." done
    • The balloon in the second paragraph of your more mathematical section would do better as a hidden section than as a mouseover. So you could use the syntax, {{HideThis|1=<text to follow "hide"/ "show">|2=<hidden text>}} to do something like this:
...And ABC is half of MBC.
We know this because the triangles we've divided the regular polygon into are congruent isosceles triangles. The base angles of an isosceles triangle are congruent, and all the triangles that make up the polygon are congruent. Therefore, the base angles of all the isosceles triangles in our regular polygon are all congruent. Because angles MBA and ABC are base angles of isosceles triangles in our polygon, they are congruent. As they are congruent and add up to angle MBC, the measure of each is half the measure of angle MBC.
So we just multiply that expression by... done
  • In your first paragraph in the more mathematical section, your perimeter measure isn't constant; make sure that, once you establish one side length as x, you always use that measure. (For example, you say "12 units" where it should be "12x units".) done
  • In the same paragraph, the jump you make from talking about hexagons to talking about a general n-gon would be easier to follow with a if you ended the paragraph at " expressed in terms of x?" Then you can begin the new paragraph, "Well, consider a polygon with n sides that are each x units long..." (Note the change to the bolded phrase; "amount of" had been awkward there.)
  • You don't need a box to direct the reader to the next image. Instead, just add put a caption under your image saying "Image 1," and you can refer directly to it by saying, "Let's say that angle MBC in Image 1 is an angle..." You can also add a link between the text and the image so that if the reader clicks "Image 1" it takes them to that picture. If you're interested in doing this, you can find the syntax in several places on one of my pages, Markus-Lyapuov Fractals. done
  • When you say, "We now have a formula for the apothem," at the end of the 5th paragraph in your more mathematical section, it's not actually immediately clear what formula you're referring to. It's clear in the image where you lay out the math, but not in the text, so I'd suggest stating the formula in the text as well.
  • I'd take final statement, "Our formula is correct" out of the box. The box makes the text smaller, but this is the exciting conclusion! We don't want it to be small! Also, just one more sentence before this statement, pointing out explicitly that the two apothem values we found are the same, would be nice here. I know it should be obvious, but there's so much math between the two values that the connection is lost a bit. And one nit-picking point: what you showed here was that the formula works, not exactly that it's correct (you showed that it was correct by deriving it), so I'd change this statement to, "Our formula works." done
But again, I really love your page. These are very minor changes. Good job! -Diana (12:56, 4/18/12)

I fixed all these things. Thanks for all your help! Emma F. 14:50, 19 April 2012 (EDT)