Solid: A three-dimensional figure. The dimensions are usually called length, width, and height (ℓ, w, and h) or height, width, and depth (h, w, and d). There are only two types of solids that appear with any frequency on the GRE: rectangular solids (including cubes) and cylinders.
Uniform solid: A solid that could be cut into congruent cross sections (parallel “slices” of equal size and shape) along a given axis. Solids you see on the GRE will almost certainly be uniform solids.
Face: The surface of a solid that lies in a particular plane. Hexagon ABCDEF is one face of the solid pictured below.
Edge: A line segment that connects adjacent faces of a solid. The sides of hexagon ABCDEF are also edges of the solid pictured above.
Base: The “bottom” face of a solid as oriented in any given diagram.
Rectangular solid: A solid with six rectangular faces. All edges meet at right angles. Examples of rectangular solids are cereal boxes, bricks, etc.
Cube: A special rectangular solid in which all edges are of equal length, e, and therefore all faces are squares. Sugar cubes and dice without rounded corners are examples of cubes.
Cylinder: A uniform solid whose horizontal cross section is a circle—for example, a soup can or a pipe that is closed at both ends. A cylinder’s measurements are generally given in terms of its radius, r, and its height, h.
Lateral surface of a cylinder: The “pipe” surface, as opposed to the circular “ends.” The lateral surface of a cylinder is unlike most other surfaces of solids that you’ll see on the GRE, first because it does not lie in a plane and second because it forms a closed loop. Think of it as the label around a soup can. If you could remove it from the can in one piece, you would have an open tube. If you then cut the label and unrolled it, it would form a rectangle with a length equal to the circumference of the circular base of the can and a height equal to that of the can.
Since a cube is a rectangular solid for which ℓ = w = h, the formula for its volume can be stated in terms of any edge:
To find the volume or surface area of a cylinder, you’ll need two pieces of information: the height of the cylinder and the radius of the base.