Two whole families of 3D solids, classified and named by their base shape — beyond just their volume formulas.
The Mnemonic
Named by the Base, Not the Whole Shape
A prism has two identical, parallel polygon bases connected by rectangular (or parallelogram) side faces. A pyramid has just ONE polygon base, with triangular side faces that all meet at a single point called the apex.
Both families are named specifically after their base shape: a triangular prism has triangle bases, a pentagonal prism has pentagon bases; a square pyramid has a square base, a hexagonal pyramid has a hexagon base. The base shape entirely determines the solid's name, regardless of how many total faces it ends up having.
Prisms have two matching parallel bases connected by rectangles. Pyramids have one base with triangular faces meeting at a single apex
💡 Memory Trick
"Two families of 3D solids and their formulas." Prism = TWO bases (think of a stack, like a book). Pyramid = ONE base, tapering to a point (think of an actual pyramid). The naming convention always describes the BASE polygon, not the number of visible faces — a "triangular prism" has triangle bases but rectangle sides, not five triangular faces.
Why It Works
Counting Faces, Edges, and Vertices Systematically
Since these families are defined structurally (not just by formula), you can predict exactly how many faces, edges, and vertices any prism or pyramid will have, just from its base's number of sides (n). A prism with an n-sided base has: n+2 faces (2 bases plus n rectangular sides), 3n edges (n on each base, plus n connecting them), and 2n vertices (n on each base).
A pyramid with an n-sided base has: n+1 faces (1 base plus n triangular sides), 2n edges (n on the base, plus n connecting to the apex), and n+1 vertices (n on the base, plus the single apex). These counting patterns follow directly and predictably from each family's defining structure — one base and an apex versus two matching bases.
Using It In A Proof
Identifying a Solid From a Description
Given a description of faces or a picture of a net, correctly classifying which family (and specific base shape) a solid belongs to is the core skill here.
1
Count the polygon bases
TWO matching parallel polygon faces → prism. ONE polygon face with everything else triangular → pyramid.
2
Identify the base shape
Count the sides of that base polygon — this determines the solid's specific name (triangular, square, pentagonal, hexagonal, etc.).
3
Use the counting formulas to verify
Cross-check your identification using the face/edge/vertex counting patterns — if the numbers don't match what the formula predicts for that base, you may have misidentified the shape.
Full Worked Example
Identifying a Solid From Its Face Count
Given: A solid has 8 faces total: 6 rectangles and 2 hexagons. Identify: the solid, and find its number of edges and vertices.
1
Identify the family
Two matching polygon bases (the hexagons) connected by rectangles → this is a prism, not a pyramid.
2
Identify the specific base
The bases are hexagons (6 sides), so this is a hexagonal prism (n = 6).
Faces = n+2 = 6+2 = 8. ✓ Matches the 8 given faces (6 rectangles + 2 hexagons), confirming the classification.
This cross-check (verifying faces = n+2 matches the given 8) confirms hexagonal prism was the right identification before finalizing the edge and vertex counts.
🎯 Quick Worked Example
A pyramid has a base with 5 sides. Find its total number of faces, edges, and vertices.
1
Find the faces. Faces = n+1 = 5+1 = 6 (1 pentagonal base + 5 triangular sides).
2
Find the edges. Edges = 2n = 2(5) = 10.
3
Find the vertices. Vertices = n+1 = 5+1 = 6 (5 base vertices + 1 apex).
📌 Exam Application
A common exam question describes a solid purely by its face shapes (e.g., "2 triangles and 3 rectangles") without naming it, and asks you to identify the solid and its base — always separate out which faces are the matching PAIR (the bases) from which are the connecting side faces before naming the solid.
⚠️ Most Common Prisms and Pyramids Mistakes
Trap 1 — Naming the solid by its total face count instead of its base: A "triangular prism" is named for its triangular BASES, even though it has 5 total faces (2 triangles + 3 rectangles) — don't confuse the base shape with the total face count.
Trap 2 — Mixing up prism and pyramid edge/vertex formulas: Prisms use 3n edges and 2n vertices; pyramids use 2n edges and (n+1) vertices — these are genuinely different formulas because pyramids have a single apex where a prism has an entire second base.
✓ Quick Self-Test
1) What structurally distinguishes a prism from a pyramid? 2) How are these solids named — by total faces, or by their base shape? 3) A prism has an octagonal (8-sided) base — find its faces, edges, and vertices. 4) A pyramid has a square base — find its faces, edges, and vertices. 5) A solid has 1 pentagon face and 5 triangle faces — what is it, and how many edges does it have?