A REGULAR HEXAGONAL PYRAMID HAS A SIDE LENGTH OF 3 INCHES A SLANT HEIGHT OF 6 INCHES AND a = 24 SQUARE INCHES. WHAT IS THE APPROXIMATE AREA SURFACE

To calculate the approximate surface area of a regular hexagonal pyramid, we need to find the area of the six triangular faces and the area of the hexagonal base, and then add them together.

First, let's find the area of each triangular face using the formula:

Area = 0.5 * base * height

In this case, the base is the side length of the hexagon, which is 3 inches, and the height is the slant height of the triangular face, which is 6 inches.

Area of a triangular face = 0.5 * 3 inches * 6 inches = 9 square inches

Since the hexagonal pyramid has 6 triangular faces, the total area of the triangular faces is 6 times the area of one triangular face:

Total area of triangular faces = 6 triangular faces * 9 square inches = 54 square inches.

Next, let's find the area of the hexagonal base. The area of a regular hexagon with side length (a) is given by the formula:

Area of a hexagon = (3√3/2) * a^2

In this case, the side length (a) of the hexagon is 3 inches:

Area of the hexagonal base = (3√3/2) * (3 inches)^2

Area of the hexagonal base ≈ 7.794

Now we can calculate the total surface area of the regular hexagonal pyramid:

Total surface area ≈ 54 square inches (triangular faces) + 7.794 square inches (hexagonal base)

Total surface area ≈ 61.794 square inches

Therefore, the approximate surface area of the regular hexagonal pyramid is approximately 61.794 square inches.