What is the formula for the pentagonal prism?

Answer 1

I'm supposing this question is referring to volume.
In which case, it's dependent upon whether you know the area of the base or not.
If you do, then it is as simple as
V=Ah
Where V = Volume, A = Base area, h = Prism height
If you don't know the area of the base, then we'll need to know the apothem height and average base vertex length.

Say you have a perfect pentagonal prism, where all base vertex lengths are the same, then good, you can just use that one length as your L variable, which will represent base vertex length.

If you do have a perfect pentagonal prism, skip this paragraph. If you have a pentagon where three sides are 3cm and two are 5.5cm, then you have to average them. 3 + 3 + 3 + 5.5 + 5.5 = 20. 20 / 5 = 4. Your average base vertex length is 4.

Now that you have your vertex length you must find the apothem height. If the pentagonal prism you're working with is a tangible object or direct scale drawing, you'll be able to locate the apothem height. To do this, find the center of one of the base vertices. Draw a line or scribe a mark from the center of the vertex directly out at a 90 degree angle. Do the same with another vertex. The point where the two apothems that you've drawn meet is the center. Now, find the length of the apothem, which will now be represented by S.
Remember, L = Base vertex length, S = Apothem height, h = Prism height.
Now, think of the prism as 5 triangles stuck together like slices of a pie. So, take the apothem height times the prism height times the base vertex length.
L * S * h
This is dividing the length of the prism like the pie I talked about in the last paragraph. We now have 5 triangular prisms. So to find this volume you would do...
B * H * h / 2
Where B = Base vertex length, H = Height of triangle from bottom vertex to the highest point, h = Prism height.
Notice I left out the / 2 in the L * S * h formula. That's the only step left for the triangular slices of prism.
L * S * h / 2
Now multiply your slices by five for the total prism pie!
Pentagonal Prism Volume = 5 * (Vertex * Apothem * Height) / 2

Hope I helped! I learned a bit while writing this too, so thanks to you I guess!