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

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

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

2. ## Re: express area of equilateral triangle as function of one side s???

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*}.

3. ## 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.

4. ## Re: express area of equilateral triangle as function of one side s???

Hello, sluggerbroth!

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

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

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

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

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