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May 17th, 2011, 08:45 PM  #1 
Newbie Joined: May 2011 Posts: 22 Thanks: 0  Height equilateral triangle  formula
Hi, why is the height of an equilateral triangle equal to (s*sqrt3)/2? I understand you could break the triangle in two, leaving a 306090 triangle. Then the relative amounts are 1:sqrt3:2, and thus the attached formula results. The height becomes: h= 2^2  (sqrt3)^2, but how does the above formula result? Thanks for explaining in steps! 
May 17th, 2011, 08:55 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: Height equilateral triangle  formula
sin(60) = h/s, where h is height and s is side length. Since sin(60) = ?(3)/2, h = ?(3)s/2. (you want the positive root). 
May 17th, 2011, 09:55 PM  #3 
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Using Pythagorean theorem, h² + (s/2)² = s² i.e. h = ?(s²  (s/2)²) = ?(s²  s²/4) = ?(3s²/4) = ?(3s²)/2 = ?3s/2.

May 18th, 2011, 08:30 PM  #4 
Newbie Joined: May 2011 Posts: 22 Thanks: 0  Re: Height equilateral triangle  formula
Hi, how do you come up with h^2+(s/2)^2 = s^2? The 306090 triangle has thee relative amounts are 1:sqrt3:2. The height becomes: h= 2^2  (sqrt3)^2, but how does the above formula result? How do you come from s^2(s^2/4) to S^3/4? Thanks 
May 18th, 2011, 08:41 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Height equilateral triangle  formula
Take an equilateral triangle having side lengths s and orient it such that the bottom edge is horizontal. Now, from the top vertex drop a vertical line down to the bottom edge, bisecting the triangle. You now have two 30°60°90° triangles. Take one of these triangles and observe that the side opposite the 30° angle is s/2, the side opposite the 60° angle call h and the side opposite the 90° angle is the hypotenuse s. Now apply the Pythagorean theorem: Taking the positive root, we have: The method of greg1313 is easier though. Taking the sine of 60°, we have: 

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