The area of an equilateral triangle is found using the following formula.

\(\displaystyle A=\frac{s^{2}\sqrt{3}}{4}\) where \(\displaystyle s=\sqrt{3}\)

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

\(\displaystyle A=\frac{3\sqrt{3}}{4}\)

If the area of ABCD is equal to 45, then the perimeter of EFG is equal to x * 1.5 = 45. 45 / 1.5 = 30, so the perimeter of EFG is equal to 30. This means that each side is equal to 10.

The height of the equilateral triangle EFG creates two 30-60-90 triangles, each with a hypotenuse of 10 and a short side equal to 5. We know that the long side of 30-60-90 triangle (here the height of EFG) is equal to √3 times the short side, or 5√3.

We then apply the formula for the area of a triangle, which is 1/2 * b * h. We get 1/2 * 10 * 5√3 = 5 * 5√3 = 25√3.

In general, the height of an equilateral triangle is equal to √3 / 2 times a side of the equilateral triangle. The area of an equilateral triangle is equal to 1/2 * √3s/ 2 * s = √3s²/4.

An equilateral triangle has three congruent sides and results in three congruent angles. This figure results in two special right triangles back to back: 30° – 60° – 90° giving sides of x - x √3 – 2x in general. The height of the triangle is the x √3 side.  So A_triangle = 1/2 bh = 1/2 * 12 * 6√3 = 36√3 cm².

To calculate the height, the length of a perpendicular bisector must be determined. If a perpendicular bisector is drawn in an equilateral triangle, the triangle is divided in half, and each half is a congruent 30-60-90 right triangle. This type of triangle follows the equation below.

\(\displaystyle a^2+b^2=c^2\rightarrow (a)^2+(a\sqrt{3})^2=(2a)^2\)

The length of the hypotenuse will be one side of the equilateral triangle.

\(\displaystyle 2a=10\).

The side of the equilateral triangle that represents the height of the triangle will have a length of \(\displaystyle \small a\sqrt{3}\) because it will be opposite the 60^o angle.

\(\displaystyle a=5\rightarrow a\sqrt{3}=5\sqrt{3}\)

To calculate the area of the triangle, multiply the base (one side of the equilateral triangle) and the height (the perpendicular bisector) and divide by two.

\(\displaystyle A=\frac{1}{2}bh=\frac{1}{2}(10)(5\sqrt3)=25\sqrt3\)

In order to find the area of the triangle, we must first calculate the height of its altitude.  An altitude slices an equilateral triangle into two \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles. These triangles follow a side-length pattern. The smallest of the two legs equals \(\displaystyle x\) and the hypotenuse equals \(\displaystyle 2x\). By way of the Pythagorean Theorem, the longest leg or \(\displaystyle h=x\sqrt{3}\).

Therefore, we can find the height of the altitude of this triangle by designating a value for \(\displaystyle x\). The hypotenuse of one of the \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) is also the side of the original equilateral triangle.  Therefore, one can say that \(\displaystyle 2x=s=2\sqrt{3}\) and \(\displaystyle x=\sqrt{3}\).

\(\displaystyle h=x\sqrt{3}\)

\(\displaystyle h=\sqrt{3} * \sqrt{3}\)

\(\displaystyle \rightarrow3\)

Now, we can calculate the area of the triangle via the formula \(\displaystyle A=\frac{1}{2}bh\).

\(\displaystyle A= \frac{1}{2} * 2\sqrt{3} * 3\)

\(\displaystyle A=3\sqrt{3}\)

\(\displaystyle \rightarrow 5.20\)

In order to find the area of the triangle, we must first calculate the height of its altitude.  An altitude slices an equilateral triangle into two \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles. These triangles follow a side-length pattern. The smallest of the two legs equals \(\displaystyle x\)and the hypotenuse equals \(\displaystyle 2x\). By way of the Pythagorean Theorem, the longest leg or \(\displaystyle h=x\sqrt{3}\).

Therefore, we can find the height of the altitude of this triangle by designating a value for \(\displaystyle x\). The hypotenuse of one of the \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) is also the side of the original equilateral triangle.  Therefore, one can say

that \(\displaystyle 2x=s=89cm\) and \(\displaystyle x=44.5cm\).

\(\displaystyle h=x\sqrt{3}\)

\(\displaystyle h=44.5cm*\sqrt{3}\)

Now, we can calculate the area of the triangle via the formula

 \(\displaystyle A=\frac{1}{2}bh\)

\(\displaystyle A= \frac{1}{2} * 89cm * 44.5cm*\sqrt{3}\)

\(\displaystyle A=3430 cm^{2}\)

Now convert to meters.

\(\displaystyle \rightarrow \frac{3430cm^{2}}{1}*\frac{1m}{100cm}*\frac{1m}{100cm}= 0.343m^{2}\)

The formula for the area of a right triangle is \(\displaystyle A = \frac{1}{2}bh\), where \(\displaystyle b\) is the length of the triangle's base and \(\displaystyle h\) is its height. Since the longest side is the hypotenuse, use the two smaller numbers given as sides for the base and height in the equation to calculate the area of Triangle A:

\(\displaystyle A = \frac{1}{2}(6)(8)\)

 \(\displaystyle A = 24\)

The formula for the area of an equilateral triangle is \(\displaystyle A = \frac{s^2\sqrt{3}}{4}\), where \(\displaystyle s\) is the length of each side. (Alternatively, you can divide the equilateral triangle into two right triangles and find the area of each). Triangle B's area is thus calculated as:

\(\displaystyle A = \frac{(8)^2\sqrt{3}}{4}\)

\(\displaystyle A = \frac{64\sqrt{3}}{4}\)

\(\displaystyle A = 16\sqrt{3}\) 

To determine which of the two areas is greater without using a calculator, rewrite the areas of the two triangles with comparable factors. Triangle A's area can be expressed as \(\displaystyle 24= 8\sqrt{3}\sqrt{3}\), and Triangle B's area can be expressed as \(\displaystyle 16\sqrt{3}= (8)(2)\sqrt{3}\). Since \(\displaystyle 2\) is greater than \(\displaystyle \sqrt{3}\), the product of the factors of Triangle B's area will be greater than the product of the factors of Triangle A's, so Triangle B has the greater area.

Since the area of a triangle is 

\(\displaystyle Area=\frac{1}{2}bh\)

you need to find the height of the triangle first. Because of the 30-60-90 relationship, you can determine that the height is \(\displaystyle 5.5\sqrt{3}\).

Then, multiply that by the base (11).

Finally, divide it by two to get 52.4.

The formula for the area of an equilateral triangle is:

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

Where \(\displaystyle s\) is the length of the side

Plugging in our values, we get:

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

\(\displaystyle A=\frac{(20m)^2\sqrt{3}}{4}=\frac{400m^2\sqrt{3}}{4}=100\sqrt{3}m^2\)

The formula for the area of an equilateral triangle is:

\(\displaystyle A=\frac{s^2\sqrt{3}}{4}\),

where \(\displaystyle s\) is the length of the sides.

Plugging in our value, we get:

\(\displaystyle A=\frac{(8cm)^2\sqrt{3}}{4}=\frac{64cm^2\sqrt{3}}{4}=16cm^2\sqrt{3}\)