Three-dimensional shapes are solid figures that have length, width, and height. In this topic, we look at several important families of 3D shapes, including prisms, pyramids, and Platonic solids. Instead of only naming shapes, we focus on their structure: their faces, edges, and vertices.

A prism has two matching, parallel bases connected by rectangular faces. The base shape gives the prism its name, such as a triangular prism, rectangular prism, or pentagonal prism. A pyramid has one base and triangular faces that meet at a single vertex. Like prisms, pyramids are named by the shape of their base, such as a square pyramid or hexagonal pyramid.

We will also discuss Platonic solids, which are special polyhedra made of congruent regular polygon faces, with the same number of faces meeting at each vertex. These include the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

As future teachers, it is important to help students classify 3D shapes by their attributes, not just by appearance. We will count vertices, edges, and faces for different prisms and pyramids, then look for patterns that allow us to generalize to any prism or pyramid with an n-sided base.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Define prism, pyramid, polyhedron, and Platonic solid.

  • Classify three-dimensional shapes based on their attributes.

  • Identify faces, edges, and vertices of 3D shapes.

  • Name prisms and pyramids based on the shape of their base.

  • Count vertices, edges, and faces for prisms and pyramids.

  • Generalize formulas for an n-gon prism and an n-gon pyramid.

  • Explain how 2D polygon ideas connect to 3D shape classification.

Key Vocabulary

  • Three-Dimensional Shape - A solid figure with length, width, and height.

  • Polyhedron - A 3D shape made of flat polygon faces.

  • Face - A flat surface of a 3D shape.

  • Edge - A line segment where two faces meet.

  • Vertex - A corner where edges meet.

  • Prism - A polyhedron with two matching, parallel bases connected by rectangular faces.

  • Pyramid - A polyhedron with one base and triangular faces that meet at one vertex.

  • Base - The polygon used to name a prism or pyramid.

  • Platonic Solid - A highly regular polyhedron made from congruent regular polygon faces.

Prism and Pyramid Comparison

  • Prism - 2 matching, parallel bases - Named by the shape of the base

  • Pyramid - 1 base - Named by the shape of the base

General Patterns

For a prism with an n-sided polygon base:

  • Vertices - 2n - There are n vertices on each of the two bases.

  • Edges - 3n - There are n edges on each base, plus n connecting edges.

  • Faces - n + 2 - There are 2 bases and n side faces.

For a pyramid with an n-sided polygon base:

  • Vertices - n + 1 - There are n vertices on the base, plus 1 top vertex.

  • Edges - 2n - There are n base edges and n edges connecting to the top vertex.

  • Faces - n + 1 - There is 1 base and n triangular faces.

Platonic Solids

  • Tetrahedron - 4 Equilateral triangle faces

  • Cube - 6 Square faces

  • Octahedron - 8 Equilateral triangle faces

  • Dodecahedron - 12Regular pentagon faces

  • Icosahedron - 20 Equilateral triangle faces

Platonic solids are special because every face is the same regular polygon, and the same number of faces meet at every vertex.

Teacher Connection

Three-dimensional geometry helps students connect visual thinking, spatial reasoning, and precise mathematical language. Students often recognize common objects, such as boxes or pyramids, before they can describe them mathematically. Teachers can help students move from “it looks like a box” to “it is a rectangular prism because it has two matching, parallel rectangular bases and rectangular side faces.”

Hands-on models are especially helpful for this topic. Students can build prisms and pyramids with nets, straws, clay, blocks, or paper models. As they count faces, edges, and vertices, they can look for patterns instead of memorizing each shape separately.

Helpful teacher questions include:

  • What shape is the base?

  • How many bases does this solid have?

  • Are the faces flat or curved?

  • Where are the edges?

  • How many vertices are on the base?

  • How does the number of sides in the base help us count the faces, edges, and vertices?

  • Is this shape a prism, a pyramid, a Platonic solid, or something else?

Quick Reflection Question

Why is it useful for students to build or handle 3D models when learning about faces, edges, and vertices? What can they notice with a physical model that might be harder to see in a flat picture?