Question

what is the area of an equilateral triangle with perimeter 12?

Answer 1

4sqrt(3)

Answer 2

2*

Answer 3

^delete

Answer 4

If the triangle is equilateral, it has three equal sides. The length of each side is then,
\[12/3=4\]units.

The area of any plane triangle is \[\frac{1}{2}(base)(perpendicular.height)\](perpendicular height is perpendicular distance from base to top). You have the base. You now need the perpendicular height.

If you draw an equilateral triangle and drop a line from the apex to the bottom, such that the line is perpendicular to the base, you'll have two congruent, right-angled triangles.

The base of each of these triangles will be 2 and the hypotenuse, 4. You need to use Pythagoras' Theorem to solve for the height:\[4^2=2^2+h^2 \rightarrow 16=4+h^2 \rightarrow h^2=12 \rightarrow \]\[h=\pm \sqrt{12}=\pm 2\sqrt{3}\]We only take the positive solution since you're not having negative length here.

So now you have everything you need: base, height, and so the area is,\[A=\frac{1}{2}\times 4\times 2\sqrt{3}=4\sqrt{3}\] square units.

Answer 5

appleman, sorry, I didn't see your answer before I ploughed in!