Question

What is the area of the equilateral triangle? Round to the nearest square centimeter.

Answer 1

well, let me teach you a derivation that will lead to a certain formula you can apply to all

Answer 2

hint:
the height of your triangle is:
\[8 \times \frac{{\sqrt 3 }}{2} = ...?\]

Answer 3

say you have an equilateral triangle with all sides equal to a

Answer 4

all sides should be 8 right?

Answer 5

yes, in your case it is 8

Answer 6

since all sides are equal, we can also conclude that all angles are equal and that is 180/3 = 60 degrees

Answer 7

then what do i do

Answer 8

the requested area is given by the subsequent formula:
\[\Large A = \frac{{base \times height}}{2} = \frac{{8 \times 8 \times \frac{{\sqrt 3 }}{2}}}{2} = ...?\]

Answer 9

say we put a height, which is equal to h.

by pythagorean identity, sin60 = opposite/hypotenuse = h/a

therefore, \[\frac{ \sqrt{3} }{ 2 }a\]

we know that area of triangle = \[ = \frac{ 1 }{ 2 }bh\]

substituting h, we will get

\[A = (\frac{ 1 }{ 2 })\frac{ \sqrt{3} }{ 2 } a = \frac{ \sqrt{3}a }{ 4 }\]