An equilateral triangle has three equal sides; therefore, to find the length of each side, divide the perimeter by :

We derive the equation of base of a triangle from the area of a triangle formula:

Similarily, we can use the same formula for the height to find the length of the side of an equilateral triangle, which is given by 

, where  is the length of the height.

Therefore, the final answer is

 . 

Since we are given the radius, we should be able to find the height of the equilateral triangle. Indeed, the center of the circle is at the intersections of the heights of the triangle, and is located  away from the edge of a given height.

Therefore 5, the radius of the circle is  of the height.

Therefore, the height must be .

From here, we can use the formula for the height of the equilateral triangle , where  is the length of the height and  is the length of a side of the equilateral triangle.

Therefore, , then  is the final answer.

Let  be the length of a side of an equilateral triangle. Then the formula for the area of an equilateral triangle with side  is

So solving  

we get .

Alternative Solution:

Without knowing this formula you can still use the Pythagorean Theorem to solve this. By drawing the height of the triangle, you split the triangle into 2 right triangles of equal size. The sides are the height,  and . Letting  stand for the unknown height, we solve 

 solving for  we get

The area for any triangle is the base times the height divided by 2. So

  or .

To find the area of a traingle, we need the height and base lengths. Plug the given values into the following formula:

The area of an equilateral triangle is given by the following formula:

 , where  is the length of a side.

Since we know the length of the side, we can simply plug it in the formula and we have  or , which is the final answer.

Since we are given a radius for the circle, we should be able to find the length of the height of the equilateral triangle, indeed, the center of the circle is  of the length of the height from any vertex.

Therefore, the height is  where  is the length of the height of the triangle. Therefore .

We can now plug in this value in the formula of the height of an equilateral triangle, where  is the length of the side of the triangle.

Therefore,  or .

Now we should plug in this value into the formula for the area of an equilateral triangle  where  is the value of the area of the equilateral triangle. Therefore , which is our final answer. 

For a given equilateral triangle with a side length  and a height , the area  is 

. Plugging in the values provided:

For a given right triangle with a side length  and a height , the area  is 

. Plugging in the values provided: