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Equilateral triangle expressed as function

  1. Aug 20, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm trying to express the function of a equilateral triangle as a function of the length of a side.
    All you know is the sides are all equal


    2. Relevant equations
    I have the answer but i dont understand (how and why they got there) and they used the A of a triangle as A= 1/2(base)(height) but I thought it was just 3*L because there is three equal sides. I realize where my confusion is the area of a equilateral triangle, how do they come up with (sq(3)/4)*x it would make more sense for me, if it was 1/3 because there is three sides.


    3. The attempt at a solution
    I know a equilateral triangle has all equal sides, so my initial thought was A= L*3 and as a function wouldnt just f(a)= a*3 work?
     
  2. jcsd
  3. Aug 20, 2013 #2
    It seems that you want to express the area of an equilateral triangle as a function of its side length. First, the area of any triangle is indeed 1/2(base * height). Let x be the side length of the equilateral triangle, then the area of the triangle is A(x) = 1/2(x * height). Now you need to find the height of the triangle. You can do this by using the Pythagorean Theorem.
     
  4. Aug 20, 2013 #3

    Mark44

    Staff: Mentor

    You are confusing the perimeter of a triangle with its area. For an equilateral triangle, the perimeter is 3L. For any triangle, the area is as you show above.
    An equilateral triangle is also equiangular, meaning that all three interior angles are the same. You'll need to use a bit a basic trig to find the altitude of such a triangle.
    No.
     
  5. Aug 20, 2013 #4

    Mark44

    Staff: Mentor

    Moved thread to Precalculus section.
     
  6. Aug 20, 2013 #5
    If you use the

    [tex]Area(triangle) = 0,5\cdot L^2\cdot \sin(\alpha)[/tex]

    it's easy to recognize
     
  7. Aug 24, 2013 #6
    lol I thought I figured it out. But I dont understand why we insert 1/2 in the Pythagorean theorem.

    So if y satisfies the height, why would we put y^2 + (1/2)*x^2 = x^2. I understand we solve for y which is height. Then we plug that into the area formula to solve for a(x)= B*H *1/2 and B is X so that I undertstand that.
    The pathagorem theorm is just a^2+b^2=c^2

    But why do we put 1/2 in the part of the X in the pythagorean theorem. I'm missing something and its super frustrating lol.. i know it will be like a duh moment once i figure it out.
     
    Last edited: Aug 24, 2013
  8. Aug 24, 2013 #7
    Okay I saw it... lol

    Just in case someone else is trying to figure it out too. The reason why you put a 1/2 in there is because x is the base and sense we are using Pythagorean theorem you grabbing the height which splits the equilateral triangle down the middle. making the base which we made X cut in half. There... ;) lol

    Thanks for everyone commenting and helping me figure this out. Much appreciated.
     
    Last edited: Aug 24, 2013
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