Volume and Surface Area of Prisms and Cylinders

Volume counts the cubes that fit inside a solid, and surface area counts the paint that covers it. For any prism or cylinder the volume shortcut is base area times length: find the end face, then stretch it along. Surface area is slower but honest work: list every face, area each one, then add. Keep units apart from the start. Lengths use cm, areas use square cm, volumes use cubic cm, and mixing them up ruins otherwise perfect working.

Every prism volume starts with its end face, whatever shape that is. A rectangular end gives length times width, a triangular end gives half base times height, and that area is step one. Step two multiplies by the length running between the two ends. Check that all measurements share the same unit before you multiply. Finish by asking if the answer is sane: a prism longer than a ruler but thinner than a pencil cannot hold litres.

Prisms and cylinders are the friendliest solids because their cross sections never change. Slice a prism anywhere parallel to its end and you get the same shape back, which is why base times length always works. Cylinders are just prisms with round ends, so their base area is pi times radius squared. Nets unfold these solids flat: a cylinder opens into two circles plus one rectangle that wraps around.

Slant a deck of cards and its volume stays put, and that is the whole principle. If two solids share the same height and the same cross section area at every level, their volumes match exactly. A leaning cylinder therefore holds exactly as much as a straight one with the same base and height. This is why the base times height rule survives tilting. When a volume problem looks exotic, compare slices and the answer often falls out.