Platonic Solids

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
Answer: Square pyramid
A square pyramid has one square face and four triangular faces. The faces are not all the same polygon, so it is not a Platonic solid.
2.
How many faces does an icosahedron have?
Answer: 20
An icosahedron has $20$ faces, all equilateral triangles, with 5 of them meeting at each vertex.
3.
Find the number of edges of a cube by counting the sides of its faces.
Answer: 12
1. How many faces does a cube have? 6
2. How many sides does each face (a square) have? 4
3. How many face sides is that in total? 24
4. Each edge is shared by 2 faces. How many edges are there? 12
4.
What shape are the faces of a dodecahedron?
1. a)Equilateral triangles
2. b)Regular pentagons
3. c)Regular hexagons
4. d)Squares
Answer: Regular pentagons
A dodecahedron has 12 regular pentagons, 3 at each vertex. Hexagons are impossible, because $3 \times 120° = 360°$ lies flat.
5.
A tetrahedron has 4 faces, each an equilateral triangle. How many edges does it have?
Answer: 6
$4 \times 3 = 12$ triangle sides, and each edge is shared by two faces: $12 \div 2 = 6$ edges.
6.
What is the largest number of squares that can meet at a vertex of a solid?
Answer: 3
1. How many degrees is one angle of a square? 90
2. What is the angle sum of 3 squares at a vertex? 270
3. What is the angle sum of 4 squares at a vertex? 360
4. The sum must be less than 360°. What is the largest number of squares? 3
7.
You roll a fair 8-sided die (an octahedron) numbered 1 to 8. What is the probability of rolling an even number?
Answer: 1/2
1. How many faces (possible outcomes) does the die have? 8
2. How many of the numbers 1 to 8 are even? 4
3. What is the probability as a simplified fraction? 1/2
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
Answer: Their angles add up to 360°, so they lie flat
$6 \times 60° = 360°$ , a full turn. The six triangles fill the plane around the point (like a hexagon), so there is no corner that sticks out.
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?
Answer: 324
$3 \times 108° = 324°$ , which is less than $360°$ , so the pentagons form a real corner. With 4 pentagons it would be $432°$ , too much.
10.
Show that at most 3 regular pentagons can meet at a vertex of a solid.
Answer: 3
1. What is the sum of the interior angles of a pentagon, $(5 - 2) \times 180$ ? 540
2. How many degrees is one angle of a regular pentagon? 108
3. What is the angle sum of 3 pentagons at a vertex? 324
4. What is the angle sum of 4 pentagons at a vertex? 432
5. The sum must be less than 360°. What is the largest number of pentagons? 3
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
Answer: Cube and octahedron
The cube has 8 vertices, 12 edges, 6 faces; the octahedron has 6 vertices, 12 edges, 8 faces. Vertices and faces swap and the edges stay the same, so they are duals.
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 $k$ ?
Answer: 50
$2\sqrt{3} \times 25 = 50\sqrt{3} \approx 86.6$ cm², so $k = 50$ . Check: 8 triangles of $\frac{\sqrt{3}}{4} \times 25$ each.
13.
Find the surface area of a tetrahedron with edge $a = 6$ cm. Round to one decimal place.
Answer: 62.4
1. What is $a^2$ ? 36
2. One face has area $\frac{\sqrt{3}}{4} \times 36 = k\sqrt{3}$ . What is $k$ ? 9
3. The 4 faces together have area $m\sqrt{3}$ . What is $m$ ? 36
4. Write $36\sqrt{3}$ as a decimal, rounded to one decimal place. 62.4
14.
An icosahedron has 20 triangular faces, and 5 faces meet at each vertex. Find the number of edges and the number of vertices.
Answer: 30, 12
1. How many triangle sides are there in total? 60
2. Each edge is shared by 2 faces. How many edges? 30
3. There are also 60 triangle corners, and each vertex collects 5. How many vertices? 12
4. Check: vertices - edges + faces = ? 2
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
Answer: No, because 3 faces meet at some vertices and 4 at others
This triangular bipyramid has 5 vertices: 3 faces meet at each of the two tips, 4 faces at each of the three middle vertices. The second condition fails, so it is not Platonic (the octahedron has 8 faces, not 6).
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?
Answer: 20
The icosahedron has 20 faces, so the dual has $20$ vertices. It also has 12 faces (one per icosahedron vertex) and 30 edges: it is the dodecahedron.
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.
Answer: 20, 12, 30
1. Each dodecahedron vertex becomes a face of the dual. How many faces does the dual have? 20
2. Each dodecahedron face becomes a vertex of the dual. How many vertices? 12
3. How many edges does the dual have? 30
4. Check for the dual: vertices - edges + faces = ? 2
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.
Answer: 720
1. At a cube vertex 3 squares meet. What is the defect, $360 - 3 \times 90$ ? 90
2. The cube has 8 vertices. What is the total defect? 720
3. At an icosahedron vertex 5 triangles meet. What is the defect, $360 - 5 \times 60$ ? 60
4. The icosahedron has 12 vertices. What is the total defect? 720