Step-by-step explanation:
(You might need to draw the triangle on a piece of paper to follow)
First I tried to find the height in terms of a.
If you draw the height in, the triangle is cut in half. That half triangle is a right triangle.
Using the Pythagoras theorem you can find out the height. Since you know that the hypotenuse is also a (because it's an equilateral triangle) the equation would be:
(a/2)² + h² = a²
Assuming that h = height
(It's a/2 because when you cut the triangle in half, the base of the right triangle became a/2)
This equation is simplified:
a²/4 + h² = a²
a² + 4h = 4a²
4h = 3a²
h² = 3a²/4
Now, remeber that everything is squared so it should be
h = a√3/2
Now we have the height. To get the area we need to multiply it by the base, which is a, and then divide it by 2.
a√3/2 × a ÷ 2
a²√3/2 ÷ 2
a²√3/4
And there you go. Thats the area.
