Find the total surface area of a hexagonal prism 6ft tall with a regular base measuring 9ft on each and an apothem of length 7.8ft

The hexagon is made up 6 equilateral triangles with sides of 9 ft

Area of one of them = (1/2)(9)(9)sin60°
= (81/2)(√3/2) = 81√3/4

area of whole hexagon
= 6(81√3/4) = 243√3/2

area of whole hexagonal prism
= 2(243√3/2) + 6(9)(6)
= (243√3 + 324) ft^2

your value for the apothem of 7.8 is not needed.
Btw, the exact value of the apothem is 9sin60°
= 9√3/2

To find the total surface area of a hexagonal prism, we need to calculate the area of each individual face and then sum them up.

A hexagonal prism has 2 hexagonal bases and 6 rectangular faces. Let's start by calculating the area of the hexagonal base.

The formula to calculate the area of a regular hexagon is:

Area = (3√3 * side^2) / 2

Given that the base side measures 9ft, we can substitute this value into the formula:

Area of hexagonal base = (3√3 * 9^2) / 2
= (3√3 * 81) / 2
= (243√3) / 2
= 121.5√3

Since there are 2 hexagonal bases in the prism, the total area of both bases would be:

Total area of hexagonal bases = 2 * (121.5√3)
= 243√3

Next, let's calculate the area of the rectangular faces.

The formula to calculate the area of a rectangle is:

Area = length * width

The length of the rectangular face would be equal to the perimeter of the hexagonal base, which is 6 times the length of one side:

Length of rectangular face = 6 * side
= 6 * 9
= 54ft

The width of the rectangular face would be equal to the height of the prism, which is 6ft.

Area of rectangular face = length * width
= 54 * 6
= 324ft^2

Since there are 6 rectangular faces in the prism, the total area of all the rectangular faces would be:

Total area of rectangular faces = 6 * 324
= 1944ft^2

Finally, we can calculate the total surface area by summing up the area of the hexagonal bases and the area of the rectangular faces:

Total surface area = Total area of hexagonal bases + Total area of rectangular faces
= 243√3 + 1944ft^2

Therefore, the total surface area of the hexagonal prism is approximately 243√3 + 1944 ft^2.