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Height of equilateral triangle
By Kathryn Bambrough (P767) on Tuesday, March 23, 1999 - 10:56 pm:

Hi, I am 27 and doing an open university maths level one course called Using Mathematics. I have not studied since I was 16 and therefore sometimes find that I don't know the basics, therefore my questions may seem simple to others but extremly frustrating to myself when I have no-one to ask. I am studying functions and modelling with functions. My question is an equilateral triangle of sides length 1m each angle being pi/3 radians. In the answer to one of my problems the book explains that the height of the triangle is sin(pi/3)=square root3/2. Can you explain to me how this is so?

I will be extremely grateful for your help. Thanks Kathryn

By Richard Samworth (Rjs57) on Wednesday, March 24, 1999 - 01:28 pm:

If you draw on the height, h, say, of the triangle, you can work out h by Pythagoras' theorem, since you have a right-angled triangle with sides h,1 and 1/2.
Pythagoras says 12 = h2 + (1/2)2 = h2 + 1/4
So h2 = 1 - 1/4 = 3/4
So h = sqrt(3)/2
This equals sin (pi/3) because, as you say, each angle of the original triangle is pi/3. So we can use trigonometry on the same triangle to say
sin(pi/3) = opposite/hypotenuse = h/1 = sqrt(3)/2

Hope this helps.

Richard