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G.9(D)·Geometry·Circles & PolyhedraSupporting

Faces, Edges, and Vertices of Polyhedra

Analyze the faces, edges, and vertices of polyhedra.

Students count all faces, edges, and vertices of polyhedra, develop prism-counting patterns, and use Euler's formula to check their results.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice45 minutes

Learn the concept

Name the features

A face is a flat polygonal surface. An edge is a segment where two faces meet. A vertex is a corner where edges meet. A polyhedron is bounded by polygonal faces, so a cylinder is not a polyhedron.

Count a prism systematically

A prism with an n-sided base has two bases and n lateral faces, giving F=n+2F=n+2. It has n vertices on each base, giving V=2nV=2n. Its edges consist of n on each base and n connecting the bases, giving E=3nE=3n.

Check with Euler's formula

For a convex polyhedron, V−E+F=2V-E+F=2, equivalently F+V=E+2F+V=E+2. Use this to check a count or find a missing count. A triangular prism has F=5F=5, E=9E=9, and V=6V=6, so 6−9+5=26-9+5=2.

Worked example

A hexagonal prism is drawn with some back edges hidden. Find its total number of faces, edges, and vertices.

  1. The base has n = 6 sides.
  2. Count two hexagonal bases and six lateral faces: F = 8.
  3. Count 6 + 6 + 6 edges: E = 18. Count six vertices on each base: V = 12.
  4. Check Euler's formula: 12 − 18 + 8 = 2.

8 faces, 18 edges, and 12 vertices

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

3rd Grade Math Describing 3D Shapes

Five Minute Math · Aaron Daffern, Dallas ISD

Review the vocabulary of faces, edges, and vertices before developing prism formulas and Euler checks.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. A convex polyhedron has 9 vertices and 16 edges. How many faces does it have?

    Show answer and reasoning

    9 faces

    Substitute into V−E+F=2V-E+F=2: 9−16+F=29-16+F=2, so F=9F=9.

  2. 2. A student assigns a triangular prism 4 faces, 6 edges, and 4 vertices. What is wrong?

    Show answer and reasoning

    Those are the counts of a triangular pyramid, not a triangular prism.

    A triangular prism has two triangular bases, three lateral faces, nine edges, and six vertices. Both sets satisfy Euler's formula, so the structure must also be checked.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 30 · 1 point

Which statement about a triangular prism is true?

  1. A triangular prism has 4 faces, 6 edges, and 4 vertices.
  2. A triangular prism has 3 faces, 6 edges, and 3 vertices.
  3. A triangular prism has 5 faces, 9 edges, and 6 vertices.
  4. A triangular prism has 6 faces, 11 edges, and 8 vertices.
Show STAAR answer and explanation

A triangular prism has 5 faces, 9 edges, and 6 vertices.

Two triangular bases and three lateral faces give 5 faces. Each base has 3 edges and 3 vertices, and 3 connecting edges give 9 edges and 6 vertices.

A triangular prism has 4 faces, 6 edges, and 4 vertices.: These counts describe a triangular pyramid, not a triangular prism.

A triangular prism has 3 faces, 6 edges, and 3 vertices.: This omits faces and the second set of base vertices.

A triangular prism has 6 faces, 11 edges, and 8 vertices.: A triangular prism has 6 vertices and 9 edges, not 8 vertices and 11 edges.

Ask the tutor about this question
Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Distinguish faces, edges, and vertices on a polyhedron.
  • Count the features of a prism from the number of sides of its base.
  • Use Euler's formula to check counts and find a missing count.

Instructional sequence

  1. 1

    Engage

    5 min

    Show a triangular prism beside a triangular pyramid. Ask students to identify the bases and explain why the solids have different counts.

  2. 2

    Model investigation

    12 min

    Count faces, edges, and vertices on physical prisms. Group edges into the two base boundaries and the connectors.

  3. 3

    Direct instruction

    10 min

    Generalize prism counts from an n-sided base and introduce Euler's formula for convex polyhedra.

  4. 4

    Guided practice

    12 min

    Check candidate count triples, include hidden features, and solve for missing counts. Explain why Euler's formula alone does not identify a solid.

  5. 5

    Closure

    6 min

    Complete the exit ticket with a grouped count and an Euler check.

Misconceptions to address

““Count only the faces and edges visible in the picture.””

Teacher move: A drawing may hide back faces, bottom faces, and rear edges. Count the entire solid using its structure.

““Every triangular solid has four faces.””

Teacher move: A triangular pyramid has one triangular base and an apex. A triangular prism has two triangular bases joined by three lateral faces.

““Any counts satisfying Euler's formula must describe the named solid.””

Teacher move: Euler's formula checks consistency, not identity. Also compare the base structure and the face and edge counts.