Equilateral triangle expressed as function

[brycenrg]

Homework Statement

I'm trying to express the function of a equilateral triangle as a function of the length of a side.
All you know is the sides are all equal

Homework Equations

I have the answer but i don't understand (how and why they got there) and they used the A of a triangle as A= 1/2(base)(height) but I thought it was just 3*L because there is three equal sides. I realize where my confusion is the area of a equilateral triangle, how do they come up with (sq(3)/4)*x it would make more sense for me, if it was 1/3 because there is three sides.

The Attempt at a Solution

I know a equilateral triangle has all equal sides, so my initial thought was A= L*3 and as a function wouldn't just f(a)= a*3 work?

 

[Natr0n]

It seems that you want to express the area of an equilateral triangle as a function of its side length. First, the area of any triangle is indeed 1/2(base * height). Let x be the side length of the equilateral triangle, then the area of the triangle is A(x) = 1/2(x * height). Now you need to find the height of the triangle. You can do this by using the Pythagorean Theorem.

 

[Mark44]

Homework Statement

I'm trying to express the function of a equilateral triangle as a function of the length of a side.
All you know is the sides are all equal

Homework Equations

I have the answer but i don't understand (how and why they got there) and they used the A of a triangle as A= 1/2(base)(height) but I thought it was just 3*L because there is three equal sides.

You are confusing the perimeter of a triangle with its area. For an equilateral triangle, the perimeter is 3L. For any triangle, the area is as you show above.

I realize where my confusion is the area of a equilateral triangle, how do they come up with (sq(3)/4)*x it would make more sense for me, if it was 1/3 because there is three sides.

An equilateral triangle is also equiangular, meaning that all three interior angles are the same. You'll need to use a bit a basic trig to find the altitude of such a triangle.

The Attempt at a Solution

I know a equilateral triangle has all equal sides, so my initial thought was A= L*3 and as a function wouldn't just f(a)= a*3 work?

No.

 

[Mark44]

Moved thread to Precalculus section.

 

[janhaa]

If you use the

$$Area(triangle) = 0,5\cdot L^2\cdot \sin(\alpha)$$

it's easy to recognize

 

[brycenrg]

lol I thought I figured it out. But I don't understand why we insert 1/2 in the Pythagorean theorem.

So if y satisfies the height, why would we put y^2 + (1/2)*x^2 = x^2. I understand we solve for y which is height. Then we plug that into the area formula to solve for a(x)= B*H *1/2 and B is X so that I undertstand that.
The pathagorem theorem is just a^2+b^2=c^2

But why do we put 1/2 in the part of the X in the pythagorean theorem. I'm missing something and its super frustrating lol.. i know it will be like a duh moment once i figure it out.

 

[brycenrg]

Okay I saw it... lol

Just in case someone else is trying to figure it out too. The reason why you put a 1/2 in there is because x is the base and sense we are using Pythagorean theorem you grabbing the height which splits the equilateral triangle down the middle. making the base which we made X cut in half. There... ;) lol

Thanks for everyone commenting and helping me figure this out. Much appreciated.