Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.2 .
To find the slant height of the triangular pyramid, we need to first calculate the lateral surface area of the pyramid.
The lateral surface area of a triangular pyramid can be calculated using the formula:
lateral surface area = 1/2 * perimeter of base * slant height
In this case, the lateral surface area is given as 78 in^2. Since the base of the pyramid is an equilateral triangle, the perimeter of the base can be calculated as 3 times the length of one side of the equilateral triangle.
To find the side length of the equilateral triangle, we use the formula for the area of an equilateral triangle:
Area = (√3 /4) * side^2
78 = (√3 /4) * side^2
78 = (√3 /4) * side^2
78 * 4/√3 = side^2
312/√3 = side^2
side = √(312/√3)
side ≈ 9.481
The perimeter of the base is 3 times the side length:
Perimeter = 3 * side
Perimeter = 3 * 9.481
Perimeter ≈ 28.44 in
Now we can plug in the values into the lateral surface area formula:
78 = 1/2 * 28.44 * slant height
78 = 14.22 * slant height
slant height = 78 / 14.22
slant height ≈ 5.476 in
Therefore, the slant height of the triangular pyramid is approximately 5.476 inches.
could i have a proper answer
Certainly! Here is the proper answer for finding the slant height of the triangular pyramid:
1. Given that the lateral surface area of the triangular pyramid is 78 in^2.
2. Because the base of the pyramid is made up of equilateral triangles, the lateral surface area can be calculated using the formula:
Lateral surface area = (1/2) * Base Perimeter * Slant Height
3. We know that the lateral surface area is 78 in^2.
4. The equilateral triangle's side length (s) can be found by first calculating the area of an equilateral triangle:
Area = (√3 / 4) * s^2
5. Equating the area to 78 in^2 and solving for s gives:
s = √(312/√3) ≈ 9.481 in
6. The base perimeter is 3 times the side length because it's an equilateral triangle.
Base Perimeter = 3 * 9.481 ≈ 28.44 in
7. Plug the values into the lateral surface area formula:
78 = (1/2) * 28.44 * Slant Height
Slant Height = 78 / 14.22 ≈ 5.476 in
So, the slant height of the triangular pyramid made of equilateral triangles with a surface area of 78 in^2 is approximately 5.476 inches.