To find the slant height of the triangular pyramid, we first need to determine the surface area of one of the equilateral triangles.
Since the triangle is equilateral, all sides are equal in length. Let's call this side length "s".
The formula for the surface area of an equilateral triangle is:
Area = (sqrt(3) / 4) * s^2
Given that the surface area of the triangular pyramid is 78 in^2, we can apply this formula to find the side length "s":
78 = (sqrt(3) / 4) * s^2
78 = sqrt(3) * s^2 / 4
312 = sqrt(3) * s^2
s^2 = 312 / sqrt(3)
s ≈ 10.39 inches
Now that we have found the side length "s", we can calculate the height of the equilateral triangle using the formula for an equilateral triangle:
Height = (sqrt(3) / 2) * s
Height = (sqrt(3) / 2) * 10.39
Height ≈ 9 in
Finally, to find the slant height of the triangular pyramid, we use the Pythagorean theorem on the triangle formed by the height, half the base of the equilateral triangle (5 in), and the slant height:
Slant height = sqrt((Height^2) + (5^2))
Slant height = sqrt((9^2) + (5^2))
Slant height = sqrt(81 + 25)
Slant height = sqrt(106)
Slant height ≈ 10.3 in
Therefore, the slant height of the triangular pyramid is approximately 10.3 inches.
6 in.
5 in.
Use the net to find the slant height of the triangular pyramid made of equilateral
triangles if its surface area is 78 in.².
1 answer
1 answer