Use nets to find the surface area of the triangular pyramid with an equilateral base. ( 13 in. 12 in. 14 in. )

Answer 1

To find the surface area of a triangular pyramid with an equilateral base, we first need to find the area of the base and the lateral area.

1. Start by finding the area of the equilateral base:
The formula for the area of an equilateral triangle is:
Area = (√3/4) x side^2
Area = (√3/4) x 12^2
Area = (√3/4) x 144
Area = √3 x 36
Area = 36√3 square inches

2. Next, find the lateral area:
The lateral area of a triangular pyramid is equal to the sum of the areas of the three triangular faces. Each face is a triangle with one side as the slant height of the pyramid and the other two sides as the sides of the equilateral base.
The slant height can be found using the Pythagorean theorem:
slant height = √(height^2 + (side/2)^2)
slant height = √(13^2 + 6^2)
slant height = √(169 + 36)
slant height = √205
slant height ≈ 14.32 inches

Now, find the area of each triangular face:
Area = (1/2) x base x height
Area = (1/2) x 12 x 14.32
Area ≈ 85.92 square inches

Since there are three identical faces, the total lateral area is:
Total lateral area = 3 x 85.92
Total lateral area ≈ 257.76 square inches

3. Finally, find the total surface area by adding the base area and the lateral area:
Total surface area = Base area + Lateral area
Total surface area = 36√3 + 257.76
Total surface area ≈ 433.06 square inches

Therefore, the surface area of the triangular pyramid with an equilateral base (13 in, 12 in, 14 in) is approximately 433.06 square inches.