Equilateral triangles have sides of all equal length and angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x\(\displaystyle \sqrt{3}\), and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = \(\displaystyle \frac{a\sqrt{3}}{2}\)

Given the height, we can now find the area of the triangle using the equation:
\(\displaystyle Area = \frac{1}{2}bh=\frac{a^2\sqrt{3}}{4}\)

Equilateral triangles have sides of equal length, with angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x\(\displaystyle \sqrt{3}\), and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = \(\displaystyle \frac{a\sqrt{3}}{2}\)

Given the height, we can now find the area of the triangle using the equation:
\(\displaystyle Area = \frac{1}{2}bh=\frac{a^2\sqrt{3}}{4}\)

The answer is \(\displaystyle 81\sqrt{3}\) .  

To find the area you would first need to find what the length of each side is: 54 divided by 3 is 18 for each side.  

Then you would need to draw in the altitude of the triangle in order to get its height.  Drawing this altitude will create two 30-60-90 degree triangles as shown in the picture.  The longer leg is \(\displaystyle \sqrt{3}\) times the short leg.  Thus the height is \(\displaystyle 9\sqrt{3}\). 

Next we plug in the base and the height into the formula to get 

\(\displaystyle \frac{1}{2}\cdot \left 18\right\cdot 9\sqrt{3}=81\sqrt{3}\ in^{2}\)

The formula for the area of an equilateral triangle with side length \(\displaystyle a\) is

\(\displaystyle Area=\frac{\sqrt{3}}{4}a^2\)

So, since \(\displaystyle a=2\:cm\),

\(\displaystyle Area=\frac{\sqrt{3}}{4}(2)^2\)

\(\displaystyle Area=\frac{\sqrt{3}}{4}*4=\sqrt{3}\:cm^2\)

The formula of the area of an equilateral triangle is \(\displaystyle \frac{\sqrt{3}}{4}x^2\) if \(\displaystyle x\) is a side.

Since the sides of our triangle have doubled, they have changed from \(\displaystyle a\) to \(\displaystyle 2a\). We can substitute \(\displaystyle 2a\) into the equation and solve for the triangle's new area in terms of \(\displaystyle a\):

\(\displaystyle area=\frac{\sqrt{3}}{4}x^2=\frac{\sqrt{3}}{4}(2a)^2=\frac{\sqrt{3}}{4}4a^2=\sqrt{3}a^2\)

The formula for the area of an equilateral triangle is \(\displaystyle \frac{\sqrt{3}}{4}x^2\) if \(\displaystyle x\) is the length of one of the triangle's sides.

In this problem, the length of one of the triangle's sides is being tripled, so we can substitute \(\displaystyle 3a\) into the equation for \(\displaystyle x\) and solve for the triangle's new area in terms of \(\displaystyle a\):

\(\displaystyle Area=\frac{\sqrt{3}}{4}x^2=\frac{\sqrt{3}}{4}(3a)^2=\frac{\sqrt{3}}{4}9a^2=\frac{9\sqrt{3}}{4}a^2\)

The formula for the area of an equilateral triangle is\(\displaystyle \frac{\sqrt{3}}{4}a^2\). For this problem's triangle, \(\displaystyle a=4\:cm\), so we can substitute \(\displaystyle 4\) into the equation for \(\displaystyle a\) and solve for the area of the triangle:

\(\displaystyle Area=\frac{\sqrt{3}}{4}a^2=\frac{\sqrt{3}}{4}(4)^2=\frac{\sqrt{3}}{4}16=4\sqrt{3}\:cm^2\)

At this point, we need to divide by \(\displaystyle 2\), since the problem asks for half of the triangle's area:

\(\displaystyle \frac{4\sqrt{3}}{2}=2\sqrt{3}\:cm^2\)

We know this triangle is equilateral since each of its sides has the same length, \(\displaystyle a\). The formula of the area of an equilateral triangle is \(\displaystyle \frac{\sqrt{3}}{4}x^2\) if \(\displaystyle x\) is the length of one of the triangle's sides.

Since our side length is \(\displaystyle 10\:cm\), we can substitute that value into the equation for \(\displaystyle x\) and solve for the area of the triangle: 

\(\displaystyle Area=\frac{\sqrt{3}}{4}x^2=\frac{\sqrt{3}}{4}(10)^2=\frac{\sqrt{3}}{4}100=25\sqrt{3}\:cm^2\)

Given that the sides of our equilateral triangle are each \(\displaystyle 5\:cm\) long, we can just plug the value in to the formula for the area of an equilateral triangle and solve for the area of the triangle:

\(\displaystyle Area=\frac{\sqrt{3}}{4}a^2\) if \(\displaystyle a\) is a side of the triangle.

\(\displaystyle Area=\frac{\sqrt{3}}{4}a^2=\frac{\sqrt{3}}{4}5^2=\frac{\sqrt{3}}{4}25=\frac{25\sqrt{3}}{4}\:cm^2\)

To find the area of any triangle we can use the formula 1/2 (base x height) , that is the base times the height divided by two.  It is important to remember any of the sides of an equaliateral triangle can be used as the base when the hieght is given. The area can be found by (6 x 4.5) divided by 2; which gives 13.5 square centimeters.