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How can I find the height of a triangle given the length of all three sides?

The only solution I could find was to use Heron's formula to find area then $A=\frac{1}{2}bh$ to find height. Is there an easier way to do this?

I'm going to be using this in a piece software I'm developing, so a simple solution makes things much easier.

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Is there anything special about the triangle? Is it right, isosceles, ... – Wintermute Nov 4 '13 at 22:45
A triangle has three heights, and maybe different heights. If you want an easy solution there must be something special with your triangle, otherwise Heron's formula. – nikamed Nov 4 '13 at 22:51
Your approach seems good to me. Easy to code. And it will execute fairly quickly because it doesn't involve any trig functions. – bubba Nov 4 '13 at 23:25
@mtiano Unfortunately no. – RedHatter Nov 4 '13 at 23:37
@bubba My approach isn't too bad ... its just been nagging at me that there's got to be a better way. – RedHatter Nov 4 '13 at 23:46
up vote 3 down vote accepted

Besides the solution you stated using Heron's formula, you can use the Law of Cosines to find one of the angles. The height is then easily obtained by dropping the appropriate perpendicular and using the definition of cosine.

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Hum... looks like Heron's will be faster. Thanks anyways! – RedHatter Nov 4 '13 at 23:44

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