if the height of a regular triangular pyramid base equals 6.5, the base equals 6, and the slant height equals 8. What is the surface area?
To find the surface area of the regular triangular pyramid, we need to calculate the area of the base and the area of each triangular face.
The area of the base can be found using the formula for the area of a triangle: A = (1/2) * base * height.
In this case, the base of the pyramid is 6 and the height is 6.5, so the area of the base is A_base = (1/2) * 6 * 6.5 = 19.5.
Each triangular face of the pyramid can be divided into two right-angled triangles. The slant height represents the hypotenuse of one of these triangles, with one leg being the height of the pyramid (6.5) and the other leg being half the base (6/2 = 3). We can use the Pythagorean theorem to calculate the other leg (lateral edge) of the triangle: l = sqrt(slant height^2 - height^2) = sqrt(8^2 - 6.5^2) ≈ 5.019.
Thus, the area of one triangular face is A_face = (1/2) * base * lateral edge = (1/2) * 6 * 5.019 ≈ 15.057.
Since the regular triangular pyramid has four triangular faces, the total surface area is given by:
Surface area = Area of the base + 4 * Area of a face
Surface area = 19.5 + 4 * 15.057
Surface area ≈ 19.5 + 60.228
Surface area ≈ 79.728
Therefore, the surface area of the regular triangular pyramid is approximately 79.728 square units.