Answer
$\mathrm{Perimeter:}\ \ P(a)=3a$
$\mathrm{Area:}\ \ A(a)=\frac{\sqrt{3}}{4}a^2$
Work Step by Step
In an equilateral triange, all three sides are equal $\ \ i.e,\ \ a=b=c $
We know that the perimeter is the sum of all the sides.
Perimeter $=P=a+b+c$
$=3a\ \ $ since all the sides are of equal length.
Write this formula as a function of $\ a\ $ as:
$ P(a)=3a$
If we cut the equilateral triangle into half, we will get a right triangle with its adjacent of length $\ \frac{a}{2}\ $, opposite side $\ h\ $, and hypotenuse $\ a$.
We know that the area of a triangle is half the product of its base and height, $\ \ A=\frac{a\times h}{2}$. We need the value of $\ h\ $.
Using Pythagorean's Theorem :
$a^2=(\frac{a}{2})^2+h^2$
$\Rightarrow\ h^2=a^2-\frac{a^2}{4}$
$\Rightarrow\ h^2=\frac{3a^2}{4}$
$\Rightarrow\ h=\frac{\sqrt{3}}{2}a$
So, the area will be written as:
$A=\frac{a\times\frac{\sqrt{3}}{2}a}{2}$
$\Rightarrow\ A=\frac{\sqrt{3}}{4}a^2$
Area as a function of $\ a\ $ is:
$A(a)=\frac{\sqrt{3}}{4}a^2$
