Use the web to find the inclined height of the triangular pyramid made of equilateral triangles if its surface area is 78 inches2.
One of the faces of the pyramid is formed by 4 equilateral triangles, one of them has a height of 5 in and a base of 6 in.
First, let's calculate the total surface area of one equilateral triangle. The formula for the area of an equilateral triangle is:
Area = (sqrt(3)/4) * side^2
For the given triangle:
Area = (sqrt(3)/4) * 6^2 = (sqrt(3)/4) * 36 = 9sqrt(3) in^2
Since there are 4 equilateral triangles forming one face of the pyramid, the total surface area for that face is:
Total area = 4 * 9sqrt(3) = 36sqrt(3) in^2
The total surface area for the whole pyramid is 78 in^2. Now, we can calculate the inclined height of the pyramid using the formula:
Total area = base * height / 2 + base * inclined height / 2 + 36sqrt(3)
78 = (6 * 5 / 2) + (6 * inclined height / 2) + 36sqrt(3)
78 = 15 + 3 * inclined height + 36sqrt(3)
3 * inclined height = 63 - 36sqrt(3)
inclined height = (63 - 36sqrt(3)) / 3 ≈ 7.62 in
Therefore, the inclined height of the triangular pyramid is approximately 7.62 inches.