Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Area, approximating triangles?

  1. Feb 21, 2008 #1
    Let [tex]A_n[/tex] be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle [tex]\frac{2\pi}{n}[/tex], show that [tex]A_n=\frac 1 2 \pi r^2\sin{\frac{2\pi}{n}}.[/tex]

    Ok, I drew a circle with congruent triangles inscribed in it. I assumed that it was an equilateral triangle, so it has height [tex]\frac{\sqrt{3}}{2}r[/tex].

    So far I have

    [tex]A_{triangle}=\frac 1 2 \cdot r \cdot \frac{\sqrt{3}}{2}r[/tex]


    [tex]A_{triangle}=\frac 1 2 \cdot r^2\cdot \sin{\frac{2\pi}{n}}[/tex]

    Now I'm stuck, maybe my assumption was incorrect, and I also do not know how to incorporate the fact that it is inscribed in the circle. I know I need to take it into consideration noticing that it wants me to express the answer with the area of a circle as part of the answer. Or perhaps [tex]\pi r^2[/tex] appears through substitutions?
    Last edited: Feb 21, 2008
  2. jcsd
  3. Feb 21, 2008 #2
    I remember learning a variation of this years ago in high school, this is a nifty little formula

    The first thing I see is that I'm not sure you even understood what you were being asked to prove

    Do it with a simple shape, like a hexagon(I tried an octagon myself but couldn't draw a circle worth a darn that circumscribed it :( )

    Draw the circle around it that touches every intersection on the hexagon. Now from the center of the circle, draw a line to every intersection and behold six triangles!

    Note that they won't necessarily be equilateral triangles since two sides are the radius of the circle and one's a chord(I think that's the term >_>) Isosceles always though, I think

    So what's the area of that triangle? The base is r, you need 1/2*base*height, the height you have to drop a perpendicular and find that, you need the sine of that angle...well you have the full circle broken into 6 things, so...

    Anyways that's a better way to start
    Last edited: Feb 21, 2008
  4. Feb 21, 2008 #3


    User Avatar
    Science Advisor
    Homework Helper

    1) I don't know what an equilateral triangle has to do with anything if you have n sides. 2) Your A_n approaches 0 as n approaches infinity, hence A_n is NOT the area of a polygon with n equal sides inscribed in a circle of radius r. Look, what's the area of an isosceles triangle with apex angle 2pi/n? Multiply that by n to get the total area.
  5. Feb 21, 2008 #4
    LOL, I assumed the chord was length r, hence the equilateral triangle. Ok, let me continue reading you and Dick's post. Must solve this!!!
  6. Feb 21, 2008 #5
    Right, which is basically where I went

    I googled, and as feared, his equation is wrong, I think you misread pi for n
  7. Feb 21, 2008 #6
    Confused pi for n? That is the final equation it wants though.

    [tex]A_n=\frac 1 2 \pi r^2\sin{\frac{2\pi}{n}}[/tex]

    Stewart 5th edition, page 326
  8. Feb 21, 2008 #7


    User Avatar
    Science Advisor
    Homework Helper

    If that's supposed to be the area of an n sided inscribed polygon, it's wrong. There must be a typo in "Stewart 5th edition, page 326".
  9. Feb 21, 2008 #8
    My calc 3 professor was a proofreader for math textbook solutions

    I doubt he was very good >_>

    Of course every time I've been so certain I'm right and the book's wrong I've been just ludicrously wrong, still it's not too surprising.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook