To find the height, we need to use the formula for the surface area of a regular triangular pyramid:
Surface Area = (base area) + (lateral area)
Since the base is a regular triangle, we can find the base area using the formula:
Base Area = (1/2) * base * height
We can use the Pythagorean theorem to find the height of the base triangle:
height^2 + (base/2)^2 = slant height^2
Plugging in the values given:
height^2 + (6/2)^2 = 8^2
height^2 + 3^2 = 8^2
height^2 + 9 = 64
height^2 = 55
Taking the square root of both sides:
height = √55 ≈ 7.4 ft
Now we can find the base area:
Base Area = (1/2) * 6 * 7.4
Base Area ≈ 22.2 ft²
Finally, we can plug the values into the surface area formula:
100 = 22.2 + (lateral area)
To find the lateral area, we subtract the base area from the surface area:
lateral area = 100 - 22.2
lateral area ≈ 77.8 ft²
The lateral area of a regular triangular pyramid is given by the formula:
lateral area = (perimeter of base) * (slant height) / 2
Since the base is a regular triangle, the perimeter is equal to 3 times the length of one side:
lateral area = 3 * side length * slant height / 2
We can solve for the side length:
77.8 = 3 * side length * 8 / 2
77.8 = 12 * side length
side length = 77.8 / 12
side length ≈ 6.5 ft
The height is the height of the base triangle, which we found to be approximately 7.4 ft.
Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100 ft.2 , the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth?(1 point)
Responses
9.3 ft.
9.3 ft.
57.3 ft.
57.3 ft.
6.5 ft.
6.5 ft.
372 ft.
3 answers
wrong choose a answer choice that makes the most sense
Based on the given options, the answer that makes the most sense is 9.3 ft.