What is the area of an equilateral triangle with the radius of 8 mm and an apothem of 4 mm?

I am going to assume you mean the by radius of the equilateral triangle, you mean the radius of the circumcircle of the triangle. This is the circle that passes through all 3 vertices of the triangle. This is 8

The apothem is 4 and it is the short vertical line from the center to the midpoint of an edge.

Construct a right triangle with the one side being the apothem and the hypotenuse being the radius. The length of the other side ( the missing one for which you are solving) is 1/2 the length of the side of the equilateral triangle. A picture will help you see this.

Now area will be this times the altitude of the triangle.

Or you can directly use the fact the the area is (square root of 3 )/ 4 x (the length of the side squared. )

So in this case 1/2 the length of the side is square root of 48 or 4(square root 3)

Now to find the area we square this entire quantity after we multiply it be 2 since it was half the side and multiply by (Square root of 3)/ 4.

So the side length is 2(sqrt 48) or 8(sqrt 3)

When you square this you have 192 now to finish just multiply this by (sqrt3)/4 to find area. I come up with 48(sqrt(3)) mm2

You can also use the fact that radius=side length x (sqrt3)/4 and solve for side length.

This gives you side length = 3(radius) divided by square root of 3.

Once again, you have side length is 8(sqrt 3)

One last method. Once you found side length is 8(sqrt 3) you know that height or altitude of the triangle is 12. Now use 1/2 base x height and you have the exact same answer.

Or,

The apothem of an equilateral triangle is 1/3 of the median, which is also the height of the triangle. So the height is 12 mm (4 x 3). In one of the right triangles that are formed by drawing the height, we have:

one half of the base = cot 60 degrees x height

base = 2(cot 60 degrees x height)

So that the area of the triangle is:

A = (base x height)/2

A = [2(cot 60 degrees x height)(height)]/2

A = cot 60 degrees x height2

A = cot 60 degrees x (12 mm)2

A = 83.14 mm2 (approximately)