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May 17th, 2011, 09:45 PM   #1
e81
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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 30-60-90 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!
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May 17th, 2011, 09:55 PM   #2
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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).
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May 17th, 2011, 10:55 PM   #3
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Using Pythagorean theorem, h + (s/2) = s i.e. h = ?(s - (s/2)) = ?(s - s/4) = ?(3s/4) = ?(3s)/2 = ?3s/2.
 
May 18th, 2011, 09:30 PM   #4
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Re: Height equilateral triangle - formula

Hi, how do you come up with h^2+(s/2)^2 = s^2?
The 30-60-90 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
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May 18th, 2011, 09:41 PM   #5
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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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