Results 1 to 3 of 3

Thread: Segments formed by an inscribed equilateral triangle

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    20

    Segments formed by an inscribed equilateral triangle

    Hello. This is a problem from a book I'm studying on, I can't understand the solution.

    "Find each segment formed by an inscribed equilateral triangle if the radius of the circle is 8"

    Solution
    R=8. Since $\displaystyle s=R\sqrt{3}=8\sqrt{3}$, the area of $\displaystyle \vartriangle ABC$ is $\displaystyle \tfrac{1}{4}s^{2}\sqrt{3}=48\sqrt{3}$.
    Also, area of circle O = $\displaystyle \pi R^{2}=64\pi $.
    Hence area of segment BDC = $\displaystyle \tfrac{1}{3}\left( 64\pi -48\sqrt{3} \right)$

    These are my questions:
    1.- Why is $\displaystyle s=R\sqrt{3}=8\sqrt{3}$?

    2.And why is the area of $\displaystyle \vartriangle ABC$ equal to $\displaystyle \tfrac{1}{4}s^{2}\sqrt{3}=48\sqrt{3}$? I know the area of a triangle is equal to $\displaystyle \tfrac{1}{2}bh$, how do you get the height?
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A riddle wrapped in an enigma
    masters's Avatar
    Joined
    Jan 2008
    From
    Big Stone Gap, Virginia
    Posts
    2,550
    Thanks
    15
    Awards
    1
    Inscribed equilateral triangle:

    Draw apothem from center perpendicular to a side.

    Draw radius from center to vertex of same side.

    Triangle formed is 30-60-90.

    If hypotenuse(radius) is 8, then side opposite the 30 deg. angle is 4 (apothem).

    Side opposite 60 degree angle is $\displaystyle 4\sqrt3$

    Thus, the length of a side is $\displaystyle 2(4\sqrt3)=8\sqrt3$

    Perimeter = $\displaystyle 3(8\sqrt3)=24\sqrt3$

    $\displaystyle Area=\frac{1}{2}Pa$, where P=perimeter, a=apothem

    Finally, $\displaystyle A=\frac{1}{2}(24\sqrt3)(4)=48\sqrt3$
    Attached Thumbnails Attached Thumbnails Segments formed by an inscribed equilateral triangle-circle.jpg  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2008
    Posts
    20
    Thanks a lot.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. equilateral triangle inscribed in a circle..
    Posted in the Geometry Forum
    Replies: 4
    Last Post: Feb 28th 2010, 09:52 PM
  2. Replies: 1
    Last Post: May 18th 2009, 10:31 PM
  3. Series of Circles inscribed in an equilateral Triangle
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Mar 25th 2008, 10:49 AM
  4. Equilateral triangle inscribed in a circle
    Posted in the Geometry Forum
    Replies: 3
    Last Post: Mar 18th 2008, 06:40 AM
  5. Equilateral Triangle Inscribed in Circle
    Posted in the Geometry Forum
    Replies: 2
    Last Post: Jan 23rd 2007, 02:35 AM

Search Tags


/mathhelpforum @mathhelpforum