Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.2 .
There is a triangle net with 4 triangles with a dotted line in the middle of the triangle in the center with a 5 beside the line then a 6 under the base of the center triangle
To find the slant height of a triangular pyramid, we first need to determine the side length of the equilateral triangle base. Since the pyramid is made up of equilateral triangles, each side length of the base triangle is equal to 6 inches (as indicated on the net).
Next, we find the height of the equilateral triangle base. The formula to find the height of an equilateral triangle with side length 's' is given by:
height = (sqrt(3) / 2) * s
Substitute s = 6 inches into the formula:
height = (sqrt(3) / 2) * 6
height = 3 * sqrt(3)
Since the pyramid has four identical triangular faces, the total surface area is the sum of the areas of the four equilateral triangles. The formula for the area of an equilateral triangle with side length 's' and height 'h' is given by:
area = (s * h) / 2
Substitute s = 6 inches and h = 3 * sqrt(3) inches into the formula:
area = (6 * 3 * sqrt(3)) / 2
area = 9 * sqrt(3) in^2
Since the total surface area is given as 78 in^2, we have:
4 * area of one equilateral triangle = total surface area
4 * 9 * sqrt(3) = 78
36 * sqrt(3) = 78
sqrt(3) = 78 / 36
sqrt(3) = 2.1667
Therefore, the slant height of the triangular pyramid is approximately 2.1667 inches.