Euler's Formula for Polyhedra

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) $V - E - F = 2$
Answer: $V - E + F = 2$
Vertices and faces are added, edges are subtracted: $V - E + F = 2$ . For a cube: $8 - 12 + 6 = 2$ .
2.
A cube has $8$ vertices, $12$ edges and $6$ faces. Calculate $V - E + F$ .
Answer: 2
$8 - 12 + 6 = -4 + 6 = 2$ , exactly as Euler's formula predicts.
3.
A triangular prism has $6$ vertices and $5$ faces. Use Euler's formula to find the number of edges.
Answer: 9
$6 - E + 5 = 2$ gives $11 - E = 2$ , so $E = 9$ . Check: $3$ edges on each triangle and $3$ edges joining them, $3 + 3 + 3 = 9$ .
4.
Which of these solids is NOT a polyhedron?
1. a)Octahedron
2. b)Cube
3. c)Cylinder
4. d)Square pyramid
Answer: Cylinder
A cylinder has a curved surface and circular bases, which are not polygons, so it is not a polyhedron and Euler's formula does not apply to it.
5.
A convex polyhedron has $20$ faces and $30$ edges. How many vertices does it have?
Answer: 12
$V = 2 + E - F = 2 + 30 - 20 = 12$ . Check: $12 - 30 + 20 = 2$ .
6.
Count the vertices, edges and faces of a pentagonal prism and check Euler's formula.
Answer: 2
1. How many vertices? (top pentagon plus bottom pentagon) 10
2. How many edges? (top, bottom and vertical edges) 15
3. How many faces? (2 bases plus the rectangular sides) 7
4. Calculate $V - E + F$ 2
7.
Count the vertices, edges and faces of a square pyramid and check Euler's formula.
Answer: 2
1. How many vertices? (corners of the base plus the apex) 5
2. How many edges? (around the base plus up to the apex) 8
3. How many faces? (the base plus the triangles) 5
4. Calculate $V - E + F$ 2
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) $V = 7$ , $E = 12$ , $F = 6$
Answer: $V = 6$ , $E = 10$ , $F = 6$
$6 - 10 + 6 = 2$ (a pentagonal pyramid), while $7 - 12 + 6 = 1$ , $8 - 12 + 7 = 3$ and $9 - 14 + 6 = 1$ break Euler's formula.
9.
A prism has $27$ edges. How many faces does it have?
Answer: 11
$3n = 27$ gives $n = 9$ , so $F = 9 + 2 = 11$ . Check: $V = 18$ and $18 - 27 + 11 = 2$ .
10.
A polyhedron has $4$ triangular faces and $4$ hexagonal faces. Find its number of edges and vertices.
Answer: 12
1. How many face sides are there in total? 36
2. How many edges? 18
3. How many faces? 8
4. How many vertices? ( $V = 2 + E - F$ ) 12
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) $3n$
Answer: $2n$
There are $n$ edges around the base and $n$ edges from the base corners to the apex: $n + n = 2n$ . For a square pyramid, $2 \times 4 = 8$ .
12.
A pyramid has $9$ faces. Find the number of sides of its base, its vertices and its edges, then check Euler's formula.
Answer: 2
1. How many sides does the base have? (faces $= n + 1$ ) 8
2. How many vertices? ( $n + 1$ ) 9
3. How many edges? ( $2n$ ) 16
4. Calculate $V - E + F$ 2
13.
A convex polyhedron has only triangular faces and $12$ vertices. How many faces and edges does it have?
Answer: 20
1. Each face has 3 sides and each edge belongs to 2 faces, so $E = \frac{3F}{2}$ . Then $12 - \frac{3F}{2} + F = 2$ becomes $12 - \frac{F}{2} = 2$ . What is $\frac{F}{2}$ ? 10
2. How many faces? 20
3. How many edges? ( $\frac{3F}{2}$ ) 30
4. Check: calculate $V - E + F$ 2
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 $V - E + F$ .
Answer: 0
1. How many vertices? (outer and inner corners, top and bottom) 12
2. How many faces? (top, bottom, outer sides, inner sides) 12
3. How many edges? (all faces are quadrilaterals) 24
4. Calculate $V - E + F$ 0
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
Answer: No, 15 face sides cannot be paired into edges
Five triangles have $5 \times 3 = 15$ sides. Each edge uses exactly two sides, so $E = 7.5$ , which is not a whole number. No such polyhedron exists. A square pyramid has only $4$ triangular faces plus a square.
16.
A prism has $30$ vertices. Find the number of sides of its base, its edges and its faces, then check Euler's formula.
Answer: 2
1. How many sides does the base have? ( $2n = 30$ ) 15
2. How many edges? ( $3n$ ) 45
3. How many faces? ( $n + 2$ ) 17
4. Calculate $V - E + F$ 2
17.
A convex polyhedron has $6$ square faces and $8$ hexagonal faces. How many vertices does it have?
Answer: 24
Face sides: $6 \times 4 + 8 \times 6 = 72$ , so $E = 36$ . With $F = 14$ : $V = 2 + 36 - 14 = 24$ .
18.
Count the vertices, edges and faces of a triangular pyramid (tetrahedron) and check Euler's formula.
Answer: 2
1. How many vertices? 4
2. How many edges? 6
3. How many faces? 4
4. Calculate $V - E + F$ 2