Name_________________________________________________________
11.4
Date __________
Three-Dimensional Figures
For use with Exploration 11.4
Essential Question What is the relationship between the numbers of
vertices V, edges E, and faces F of a polyhedron?
1
EXPLORATION: Analyzing a Property of Polyhedra
Work with a partner. The five Platonic solids are shown below. Each of these solids
has congruent regular polygons as faces. Complete the table by listing the numbers of
vertices, edges, and faces of each Platonic solid.
Solid
Vertices, V
Edges, E
Faces, F
tetrahedron
cube
octahedron
dodecahedron
icosahedron
2. What is the relationship between the numbers of vertices V, edges E, and faces F of a
polyhedron? (Note: Swiss mathematician Leonhard Euler (1707–1783) discovered a formula
that relates these quantities.)
______ + ______ = ______ + 2
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Geometry
Student Journal
329
Using Euler’s Formula
______ + _______ = ______ + 2
Verify Euler’s Formula for each figure.
1.
Faces = ____
Vertices = _____
Edges = _____
______ + _______ = ______ + 2
2.
Faces = ____
Vertices = _____
Edges = _____
______ + _______ = ______ + 2
Solve:
3. A polyhedron has 8 faces and 12 vertices. How many edges does it have?
4. A polyhedron has 24 vertices and 36 edges. How many faces does it have?
5. A polyhedron has 32 faces and 60 edges. How many vertices does it have?
330 Geometry
Student Journal
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All rights reserved.