The volume of the prism is the area of its base multiplied by the height. Since it is a regular pentagon, we can find the measure of each interior angle by substituting n=5 into the formula 180^(∘)(n-2)n and simplifying.
Each interior angle is 108^(∘). Let's illustrate this in a diagram.
To calculate the area of the base we can draw segments between the midpoint of the pentagon and each vertex, thereby creating 5 congruent isosceles triangles. If we obtain the area of one triangle we can find the area of the base by multiplying this value by 5.
Using the tangent ratio, we can calculate the height of the triangle.
The height of the triangle is about 4.13 feet. With this information we can find the area of the triangle and finally, if we multiply this by 5, the area of the base.
A=(1/2(6)(4.13)5 ≈ 61.95ft^2
The base has an area of about 61.95 ft^2. To find the prism's volume we multiply the area of the base by the prism's height.
V=61.95(12)≈ 743.4ft^3
To find the surface area of the prism we have to add the area of the two bases with the area of the sides. The area of one base is 61.95ft.^2, which means the area of both bases must be 123.9 ft^2. Each side is rectangular with a width of 6 feet and a length of 12 feet. Since there are 5 sides, we can find the total area of the sides by multiplying the area of one side by 5.
A=(6)(12)(5)=360 ft^2
Finally, we will add all of the external faces to get the total surface area of the prism.
123.9+360=483.9 ft^2
