Three-Dimensional Figures
A three-dimensional figure, or solid, takes up space instead of lying flat. The solids in this lesson split into two families: polyhedrons, whose surfaces are made entirely of flat polygon faces — prisms, pyramids, cubes — and solids with curved surfaces, like cylinders, cones, and spheres, which are not polyhedrons.
For any polyhedron you should be able to do two things: classify it by its base, and count its faces, edges, and vertices. A face is a flat polygon surface, an edge is the segment where two faces meet, and a vertex is the point where edges meet. Counting these carefully — including the parts hidden at the back of the drawing — is the core skill.
Prisms and pyramids
A prism has two parallel, congruent bases connected by rectangular lateral faces. The bases name the prism: triangular bases make a triangular prism, rectangular bases make a rectangular prism. The two bases sit across from each other, and the prism has the same cross-section all the way through.
A pyramid has just one base, and its lateral faces are triangles that all meet at a single vertex — the apex. Again the base does the naming: a pyramid on a square base is a square pyramid, on a triangular base a triangular pyramid.
To classify a solid, ask two questions. What shape is the base? And do the side faces run parallel to a second base (prism) or come to a point (pyramid)?
The figure shows a square pyramid — one square base with triangular faces rising to a single apex.
Solids that are not polyhedrons
Every face of a polyhedron must be a polygon — a flat shape with straight sides. A cylinder has two circular bases and a curved side, a cone has a circular base and a curved surface rising to a point, and a sphere is curved everywhere. Circles are not polygons, so none of these solids is a polyhedron, even though a cylinder's bases are flat.
Counting faces, edges, and vertices
Count in an organized order or you will miss the hidden pieces at the back of the drawing. For a prism: the two bases plus one rectangle per side of the base gives the faces; the edges around the top base, the edges around the bottom base, and the vertical edges connecting them give the edges; the corners of the two bases give the vertices.
There is a pattern worth knowing. A prism whose base has sides has faces, edges, and vertices. A pyramid whose base has sides has faces, edges, and vertices. If your counts do not fit the pattern, recount — a dashed hidden edge probably got skipped.
Worked examples
Example 1: count the parts of a rectangular prism
How many faces, edges, and vertices does a rectangular prism have?
Answer: faces, edges, vertices
Example 2: faces and edges of a triangular prism
How many faces and how many edges does a triangular prism have?
Answer: faces and edges
Example 3: classify and count a pentagonal pyramid
A pyramid has a pentagon for its base. Classify the solid and find how many faces it has.
Answer: A pentagonal pyramid with faces
Try one yourself
Common questions
What is the difference between a prism and a pyramid?
A prism has two parallel, congruent bases and keeps the same cross-section from one base to the other. A pyramid has one base, and its lateral faces are triangles that meet at a single apex. Two matching ends means prism; one base rising to a point means pyramid.
Why isn't a cylinder a polyhedron?
Every face of a polyhedron must be a polygon, made of straight segments. A cylinder's bases are circles and its side is a curved surface — neither is a polygon. Flat is not enough; the faces must have straight sides.
What exactly counts as an edge?
An edge is the segment where two faces meet. On a rectangular prism, each of the line segments outlining the box is an edge. The dashed segments in a drawing are hidden edges at the back of the solid — they still count.
Is a cube a prism?
Yes. A cube is a rectangular prism whose faces are all congruent squares. It has the standard prism counts: faces, edges, and vertices.
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