I can do the illustrations myself

  6 +2√3 ft²

Step-by-step explanation:

Given a triangular pyramid with ...

Find

Solution

Since corresponding edges are the same length, the area of each of the three faces is (6 ft²)/3 = 2 ft². This can be computed by ...

  A = (1/2)s²·sin(α)

where s is the lateral edge length and α is the angle at the apex formed by the two edges that meet there. Filling in the given values, we find ...

  2 ft² = (1/2)(2 ft)²·sin(α)

  1 = sin(α)   ⇒   α = 90°

That is, each face of the pyramid is an isosceles right triangle with legs of length 2 ft. The hypotenuse of that triangle, the base edge of the pyramid, is then 2√2 ft.

So, the base is an equilateral triangle with edge lengths 2√2 ft. Its area can be computed from ...

  A = (√3)/4·s²

where s is the edge length of the equilateral triangle. That is, the base area is ...

  A = (√3)/4·(2√2)² = 2√3 . . . . square feet

So, the total surface area of the pyramid is ...

  (6 +2√3) ft² ≈ 9.4641 ft² . . . . . total surface area