The formula for the area of an equilateral triangle is:

,

where  is the length of the sides.

Plugging in our value, we get:

The formula for the perimeter of an equilateral triangle is:

Plugging in our values, we get

The formula for the area of an equilateral triangle is:

Plugging in our values, we get

Note that an equilateral triangle has equal sides and equal angles. The question gives us the length of the base, 5, but doesn't tell us the height. 

If we split the triangle into two equal triangles, each has a base of 5/2 and a hypotenuse of 5. 

Therefore we can use the Pythagorean Theorem to solve for the height:

Now we can find the area of the triangle:

Recall that an equilateral triangle also obeys the rules of isosceles triangles.   That means that our triangle can be represented as having a height that bisects both the opposite side and the angle from which the height is "dropped."  For our triangle, this can be represented as:

Now, although we do not yet know the height, we do know from our 30-60-90 regular triangle that the side opposite the 60° angle is √3 times the length of the side across from the 30° angle. Therefore, we know that the height is 3√3.

Now, the area of a triangle is (1/2)bh. If the height is 3√3 and the base is 6, then the area is (1/2) * 6 * 3√3 = 3 * 3√3 = 9√(3).

The radius of the circle is 2, from the equation circumference . Each triangle is the same, and is equilateral, with side length of 2. The area of a triangle 

To find the height of this triangle, we must divide it down the centerline, which will make two identical 30-60-90 triangles, each with a base of 1 and a hypotenuse of 2. Since these triangles are both right traingles (they have a 90 degree angle in them), we can use the Pythagorean Theorem to solve their height, which will be identical to the height of the equilateral triangle.

We know that the hypotenuse is 2 so . That's our  solution. We know that the base is 1, and if you square 1, you get 1.

Now our formula looks like this: , so we're getting close to finding .

Let's subtract 1 from each side of that equation, in order to make things a bit simpler: 

Now let's apply the square root to each side of the equation, in order to change  into : 

Therefore, the height of our equilateral triangle is 

To find the area of our equilateral triangle, we simply have to multiply half the base by the height: 

The area of our triangle is 

The altitude, , divides the equilateral triangle into two  right triangles and divides the bottom side in half.  

In a  right triangle, the sides of the triangle equal , , and . In these equations  equals the length of the smallest side, which in our triangle is  or .  

In this scenario:

and

Therefore, 

An altitude slices an equilateral triangle into two  triangles. These triangles follow a side-length pattern. The smallest of the two legs equals  and the hypotenuse equals . By way of the Pythagorean Theorem, the longest leg or .

Therefore, we can find the height of the altitude of this triangle by designating a value to . The hypotenuse of one of the  is also the side of the original equilateral triangle.  Therefore, one can say that  and .

When you draw the height in an equilateral triangle, it makes two 30-60-90 triangles. Because of that relationship, the height (which is across from the ) is . 

Each angle in an equilateral triangle is .

Use the formula for  triangles in order to find the length of the height.

The formula is:

Where  is the length of the side opposite the 

If we were to create a  triangle by drawing the height, the length of the side is , the base is , and the height is .

If we draw a line segment between X and the base of the triangle, we form a  triangle.

We can use the relationships between the sides of a  triangle in order to find the length of X.

We know the base opposite the  is .

The value of the height opposite the must then be , or .

Therefore, the value of X will be twice the value of the height: