Results 1 to 2 of 2

Math Help - Find the length of the side of a regular octagon inscribed in a circle of radius 1

  1. #1
    Newbie
    Joined
    Jan 2013
    From
    US
    Posts
    20

    Find the length of the side of a regular octagon inscribed in a circle of radius 1

    Please don't use sine, cosine, or tangent when solving. Solve it by introducing segments, etc.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,406
    Thanks
    1294

    Re: Find the length of the side of a regular octagon inscribed in a circle of radius

    The easiest way is to realise that the octagon is made up of eight congruent isosceles triangles, each with two lengths 1/2 a unit long and with their included angle as 45 degrees.

    Then the area of each triangle is

    \displaystyle \begin{align*} A &= \frac{1}{2} ab\sin{C} \\ &= \frac{1}{2}\cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \sin{\left( 45^{\circ} \right)} \\ &= \frac{1}{8}\cdot \frac{\sqrt{2}}{2} \\ &= \frac{\sqrt{2}}{16} \end{align*}

    Therefore the entire area of the octagon is \displaystyle \begin{align*} 8 \cdot \frac{\sqrt{2}}{16} = \frac{\sqrt{2}}{2} \end{align*}.

    I'm not sure if it can be done without using some trigonometry tbh...


    Edit: I misread the question, I thought you wanted the area. The length of each segment is easily found using the Cosine Rule, since you have two sides of each triangle and their included angle.

    \displaystyle \begin{align*} c^2 &= a^2 + b^2 - 2ab\cos{(C)} \\ c^2 &= \left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^2 - 2\left( \frac{1}{2} \right)\left( \frac{1}{2} \right) \cos{\left( 45^{\circ} \right)} \\ c^2 &= \frac{1}{4} + \frac{1}{4} - \frac{1}{2} \left( \frac{\sqrt{2}}{2} \right) \\ c^2 &= \frac{2 - \sqrt{2}}{4} \\ c &= \frac{\sqrt{ 2 - \sqrt{2} }}{2} \end{align*}

    The length of each side in the octagon is \displaystyle \begin{align*} \frac{\sqrt{2 - \sqrt{2}}}{2} \end{align*} units.
    Last edited by Prove It; February 5th 2013 at 07:29 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Regular Octagon Inscribed Inside of a Square
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: September 10th 2012, 05:14 PM
  2. Replies: 1
    Last Post: May 22nd 2012, 05:22 PM
  3. Replies: 2
    Last Post: October 18th 2010, 04:32 AM
  4. Replies: 2
    Last Post: February 6th 2010, 08:31 AM
  5. Replies: 1
    Last Post: February 20th 2008, 01:00 PM

Search Tags


/mathhelpforum @mathhelpforum