Platonic Solids

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Show your work for each problem.

1.
Which of these solids is NOT a Platonic solid?
1. a)Square pyramid
2. b)Icosahedron
3. c)Cube
4. d)Octahedron
2.
How many faces does an icosahedron have?
3.
Find the number of edges of a cube by counting the sides of its faces.
4.
What shape are the faces of a dodecahedron?
1. a)Equilateral triangles
2. b)Regular pentagons
3. c)Regular hexagons
4. d)Squares
5.
A tetrahedron has 4 faces, each an equilateral triangle. How many edges does it have?
6.
What is the largest number of squares that can meet at a vertex of a solid?
7.
You roll a fair 8-sided die (an octahedron) numbered 1 to 8. What is the probability of rolling an even number?
8.
Why can 6 equilateral triangles never meet at a vertex of a Platonic solid?
1. a)The solid would have too many edges
2. b)The triangles would be too small
3. c)Euler's formula does not allow 6 faces
4. d)Their angles add up to 360°, so they lie flat
9.
At each vertex of a dodecahedron, 3 regular pentagons meet. Each angle of a regular pentagon is $108°$ . What is the angle sum at one vertex, in degrees?
10.
Show that at most 3 regular pentagons can meet at a vertex of a solid.
11.
Which pair of Platonic solids are duals (vertices and faces swap)?
1. a)Octahedron and icosahedron
2. b)Tetrahedron and cube
3. c)Cube and octahedron
4. d)Dodecahedron and cube
12.
The surface area of an octahedron is $2\sqrt{3}\,a^2$ . For edge $a = 5$ cm the area is $k\sqrt{3}$ cm². What is $k$ ?
13.
Find the surface area of a tetrahedron with edge $a = 6$ cm. Round to one decimal place.
14.
An icosahedron has 20 triangular faces, and 5 faces meet at each vertex. Find the number of edges and the number of vertices.
15.
Glue two regular tetrahedra together along one face. The new solid has 6 identical equilateral triangles as faces. Is it a Platonic solid?
1. a)Yes, because all faces are identical equilateral triangles
2. b)No, because triangles cannot form a closed solid
3. c)Yes, it is the octahedron
4. d)No, because 3 faces meet at some vertices and 4 at others
16.
Put a point at the center of each face of an icosahedron and join the points of neighboring faces to make its dual solid. How many vertices does the dual have?
17.
The dodecahedron has 20 vertices, 30 edges and 12 faces. Find the number of faces, vertices and edges of its dual, and check Euler's formula.
18.
The angle defect at a vertex is $360°$ minus the angle sum there. Find the total angle defect of the cube and of the icosahedron.