# express area of equilateral triangle as function of one side s???

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• May 2nd 2012, 07:04 AM
sluggerbroth
express area of equilateral triangle as function of one side s???
express area of equilateral triangle as function of one side s???
• May 2nd 2012, 07:17 AM
Prove It
Re: express area of equilateral triangle as function of one side s???
Quote:

Originally Posted by sluggerbroth
express area of equilateral triangle as function of one side s???

Use Heron's formula \displaystyle \begin{align*} A = \sqrt{S(S - a)(S - b)(S-c)} \end{align*} where S is the semiperimeter, so \displaystyle \begin{align*} S = \frac{a + b + c}{2} \end{align*}, and \displaystyle \begin{align*} a = b = c = s \end{align*}.
• May 2nd 2012, 07:49 AM
biffboy
Re: express area of equilateral triangle as function of one side s???
Let h = height of triangle. Then h=s*sin60 So area= 1/2*s*s*sin60=1/2*s^2*sin60 Sin60=root3/2 so area = root3*s^2/4
Can get the height by 'splitting' triangle in two and using Pythagoras, if you prefer.
• May 2nd 2012, 07:50 AM
Soroban
Re: express area of equilateral triangle as function of one side s???
Hello, sluggerbroth!

Quote:

$\text{Express area of equilateral triangle as function of one side }s.$

Code:

              A               *             /|\             / | \           /  |  \         s /  |  \ s         /    |h  \         /    |    \       /      |      \     B * - - - * - - - * C       :  s/2  D  s/2  :
$\text{The base of the triangle is }s.$
$\text{The height is }h = AD.$

$\text{In right triangle }ADC:\:h^2 + (\tfrac{s}{2})^2 \:=\:s^2 \quad\Rightarrow\quad h^2 + \tfrac{s^2}{4} \:=\:s^2$

. . $h^2 \:=\:\tfrac{3}{4}s^2 \quad\Rightarrow\quad h \:=\:\tfrac{\sqrt{3}}{2}s$

$\text{Therefore: }\:A \;=\;\tfrac{1}{2}bh \;=\;\tfrac{1}{2}(s)\left(\tfrac{\sqrt{3}}{2}s \right) \;=\;\frac{\sqrt{3}}{4}s^2$