Euler's Formula for Polyhedra

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Show your work for each problem.

1.
Which equation is Euler's formula for a convex polyhedron?
1. a) $V + E + F = 2$
2. b) $V - E + F = 2$
3. c) $V + E - F = 2$
4. d) $V - E - F = 2$
2.
A cube has $8$ vertices, $12$ edges and $6$ faces. Calculate $V - E + F$ .
3.
A triangular prism has $6$ vertices and $5$ faces. Use Euler's formula to find the number of edges.
4.
Which of these solids is NOT a polyhedron?
1. a)Octahedron
2. b)Cube
3. c)Cylinder
4. d)Square pyramid
5.
A convex polyhedron has $20$ faces and $30$ edges. How many vertices does it have?
6.
Count the vertices, edges and faces of a pentagonal prism and check Euler's formula.
7.
Count the vertices, edges and faces of a square pyramid and check Euler's formula.
8.
Which set of counts can belong to a convex polyhedron?
1. a) $V = 8$ , $E = 12$ , $F = 7$
2. b) $V = 9$ , $E = 14$ , $F = 6$
3. c) $V = 6$ , $E = 10$ , $F = 6$
4. d) $V = 7$ , $E = 12$ , $F = 6$
9.
A prism has $27$ edges. How many faces does it have?
10.
A polyhedron has $4$ triangular faces and $4$ hexagonal faces. Find its number of edges and vertices.
11.
A pyramid has a base with $n$ sides. How many edges does it have?
1. a) $n + 1$
2. b) $2n$
3. c) $n + 2$
4. d) $3n$
12.
A pyramid has $9$ faces. Find the number of sides of its base, its vertices and its edges, then check Euler's formula.
13.
A convex polyhedron has only triangular faces and $12$ vertices. How many faces and edges does it have?
14.
A triangular picture frame is a solid with a triangular hole through it. Its top and bottom are each split into $3$ trapezoids, so every face is a quadrilateral. Find $V - E + F$ .
15.
Can a polyhedron have exactly $5$ faces that are all triangles?
1. a)Yes, it has 7 edges
2. b)No, 15 face sides cannot be paired into edges
3. c)Yes, it is a square pyramid
4. d)No, it would need 0 vertices
16.
A prism has $30$ vertices. Find the number of sides of its base, its edges and its faces, then check Euler's formula.
17.
A convex polyhedron has $6$ square faces and $8$ hexagonal faces. How many vertices does it have?
18.
Count the vertices, edges and faces of a triangular pyramid (tetrahedron) and check Euler's formula.