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11.6 Polyhedron Faces, Edges,

and Vertices

GOAL

· pipe cleaners or straws · modelling clay

Determine how the number of faces, edges, and vertices of a polyhedron are related.

Learn about the Math

Toma and Benjamin noticed that whenever you make a 2-D polygon, the number of vertices is the same as the number of edges. They wondered whether the number of faces, vertices, and edges of 3-D polyhedrons are related.

links the number of faces, edges, and ? What patternpolyhedron? vertices of a

Benjamin tried building an unusual polyhedron first. "I'll start building it from the top. The number of vertices and number of edges are the same, and there is one face."

Part built top Number of faces 1 Number of vertices 4 Number of edges 4

390 Chapter 11

NEL

"Next, I'll add squares. For each square, I add one new face, two new vertices, and three new edges. The total number of new faces and new vertices is equal to the number of new edges. "Now I'll add triangles. For each triangle, I add one new face and one new edge, but no new vertices. Again, the total number of new faces and vertices is the same as the number of new edges."

Part built top 4 new squares 4 new triangles Number of faces 1 4 more 4 more Number of vertices 4 8 more 0 more Number of edges 4 12 more 4 more

A. Construct the parts that Benjamin constructed. Add the next set of squares. Explain why the number of edges is 1 less than the total number of faces and vertices. B. Add the bottom of the polyhedron. Explain why you have added no new edges or vertices, but one new face. C. Why is the number of edges 2 less than the total number of faces and vertices? D. Choose one of the following shapes. Compare the number of edges with the total number of faces and vertices. What do you observe?

E. Compare your results with the results of students who chose different shapes. What do you notice?

Reflecting

1. The relationship you described in step C is called Euler's formula (pronouced "oiler"). Explain why it can be written as F V E 2, where F is the number of faces, V is the number of vertices, and E is the number of edges of the shape. 2. How does Euler's formula allow you to predict the number of edges, faces, or vertices of a shape if you know two of these values?

NEL

Geometry and Measurement Relationships

391

Work with the Math

Example 1: Checking whether a polyhedron is possible

Is it possible to make a polyhedron with 6 faces, 7 vertices, and 10 edges?

Tran's Solution

F V E 2 I used Euler's formula. If it is possible to make a polyhedron like this, the result should be 2 when I substitute the values into Euler's formula. I substituted the values into the formula. The result is 3, not 2, so it is not possible to make such a polyhedron.

F

V

E

6 3

7

10

Example 2: Using Euler's formula to determine a missing value

If a polyhedron has 10 faces and 18 edges, how many vertices should it have?

Benjamin's Solution

F V E 10 V 18 V 8 8 8 V 2 2 2 2 8 10 I used Euler's formula. I substituted 10 for the number of faces and 18 for the number of edges.

V

I used balancing to solve the equation. The polyhedron should have 10 vertices.

A

Checking

B

Practising

3. A student used 10 pipe cleaners to make the edges of a polyhedron. If the polyhedron has 6 vertices, how many faces must it have? 4. Show that Euler's formula works for a tetrahedron.

5. Show that Euler's formula works for the other four Platonic solids: a cube, an octahedron, a dodecahedron, and an icosahedron.

392 Chapter 11

NEL

6. Copy and complete the chart for some polyhedrons.

Number of faces 6 6 20 16 Number of edges 9 12 30 12 10 6 Number of vertices 5 7

10. Make another cube using modelling clay. Then make a pyramid on each face of the cube. Show that Euler's formula works for this polyhedron. 11. Imagine that you drilled a rectangular hole through a cube. Does Euler's formula work for the new shape? 12. a) Construct a triangular prism.

7. The following crystals and gemstones have been cut to form polyhedrons. Show that Euler's formula works for each polyhedron.

b) c) d) e)

a)

How many faces does the prism have? How many edges does the prism have? How many vertices does the prism have? Show that Euler's formula works for the prism.

13. Repeat question 12 using a pentagonal pyramid.

C

Extending

b)

14. Make a cube using modelling clay. Mark a point in the centre of each face. Imagine that you joined these points with string inside the cube to form a polyhedron. Show that Euler's formula works for this polyhedron. 15. A prism has a base with n sides.

8. Show that Euler's formula works for this cuboctahedron.

a) b) c) d)

How many faces does the prism have? How many edges does the prism have? How many vertices does the prism have? Show that Euler's formula works for the prism.

16. A pyramid has a base with n sides.

9. Make a cube using modelling clay. Cut the corners off the cube. Show that Euler's formula works for the new shape.

NEL

a) How many faces does the pyramid have? b) How many edges does the pyramid have? c) How many vertices does the pyramid have? d) Show that Euler's formula works for the pyramid.

Geometry and Measurement Relationships

393

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