Surface Area and Volume of Prisms, Cylinders, Cones, and Spheres

Every solid has a net: unfold it flat and the curved or boxed-in shape becomes a collection of rectangles, circles, and triangles you already know how to measure. A cylinder unfolds into two circles plus one rectangle, where the rectangle's width is the circle's circumference. A rectangular prism unfolds into three pairs of matching rectangles. Add those flat pieces and you have the surface area. Volume works differently: for a prism or cylinder it's base area times height, because you're stacking identical flat layers from the bottom to the top. A cone or pyramid only fills a third of the prism or cylinder with the same base and height, because the solid tapers to a point instead of keeping full width the whole way up — a fact you can see by packing three cones' worth of sand into one matching cylinder. A sphere's volume, 4/3 π r³, comes from adding up infinitely many thin cone-shaped slices from the center out to the surface, each with height r.

These formulas sit next to each other on the test because the same figure needs both: one question might give you a cylinder's dimensions and ask for volume, then reuse the same cylinder and ask for surface area, and mixing up which formula goes with which word costs easy points. Area formulas always carry squared units; volume formulas always carry cubed units — checking units catches most mistakes before you even finish the arithmetic.