Exercises: Polyhedra

An architect is designing a new glass display for a shopping centre in the shape of a right pentagonal prism. Look at the drawing and answer the following questions to help them with the material inventory:

1. a) How many faces does the prism have in total?
2. b) How many vertices does the figure have?
3. c) How many edges are there in total?

In a Design and Technology class, students are building a structure for a model. They have designed a polyhedron with $7$ faces and $10$ vertices. Use Euler's formula ($C + V = A + 2$) to calculate how many rods (edges) they will need to assemble the complete structure.

The dice used in role-playing games usually have the shapes of regular polyhedra. If you are playing with a die shaped like an octahedron (look at the figure, where the hidden edges are shown dashed):

1. a) What shape are its faces?

1. b) How many faces does this die have?

1. c) How many edges meet at each of its vertices?

Classify the following pyramids according to the polygon of their base and determine whether they are regular or irregular, justifying your answer:

1. a) A pyramid whose base is a square with sides of $5\text{ cm}$ and whose lateral faces are congruent isosceles triangles.
2. b) A pyramid whose base is a rectangle measuring $4\text{ cm}$ by $6\text{ cm}$.

Aoife wants to make a cardboard box for a gift in the shape of a cube. To do this, she draws different templates in her notebook. Explain which of the following two nets can form a perfect cube and which cannot.

You can do it on paper and send a photo with your answer.

Look around your classroom or room and name:

1. a) An object shaped like a cuboid.
2. b) How many faces does that object have?
3. c) Are all its faces identical or are there different faces? Describe what they are like.

State which of the following statements about polyhedra is FALSE and explain why:

1. A convex polyhedron is one that, when resting on any face, does not "cave in".

2. The regular tetrahedron is the polyhedron with the minimum possible number of faces.

3. In a regular polyhedron, a different number of edges can meet at each vertex.

4. The cube is a special type of square prism.

Calculate the total surface area of a Rubik's cube if the length of each of its edges is $6\text{ cm}$. Remember that a cube has $6$ identical square faces.

Lucas is designing a regular octagonal pyramid model in Technology class. In his notebook, he has written: «Since the base is an octagon, the pyramid has $8$ faces in total, $8$ vertices, and $8 \times 2 = 16$ edges. Furthermore, Euler's formula holds because $8 + 8 = 16$». Identify the three conceptual errors that Lucas made, explain with reasoning how the elements of a pyramid are actually counted from the sides of its base, and obtain the correct values for faces ($C$), vertices ($V$), and edges ($A$), verifying Euler's Theorem with them.

For a geometry exhibition at your secondary school, you have to classify several wooden pieces according to their properties. You have a piece that is a polyhedron and you know it has exactly $18$ edges and $8$ faces, but you are unsure whether it is a prism or a pyramid. Determine with reasoned justification what type of geometric body it is and what polygon forms its base. Next, answer with justification: would it be possible to build a pyramid that has exactly $15$ edges in total?

A convex polyhedron has the same number of faces as a regular octahedron and the same number of vertices as the number of edges of a cube. Use Euler's formula to determine how many edges this geometric solid has. Schematically represent a polyhedron that satisfies these characteristics (for example, a hexagonal prism).

A well-known chocolate brand sells its products in boxes shaped like a regular triangular prism. If the side length of the base triangle is $6 \text{ cm}$ and the height of the prism is $20 \text{ cm}$, calculate the total surface area of cardboard needed to make one box. (Remember that you must first find the height of the base triangle using Pythagoras' theorem).

Calculate the volume of a regular square pyramid whose base edge measures $10 \text{ cm}$ and whose slant edge measures $13 \text{ cm}$. To solve this, you will need to calculate the apothem of the pyramid first and then its height using Pythagoras' theorem.

Un escultor quiere crear una pieza tallando un cubo de madera de $10 \text{ cm}$ de arista. Si le quita las cuatro esquinas superiores cortando por los puntos medios de las aristas que concurren en cada vértice (en la figura, los cortes aparecen en naranja), cuenta cuidadosamente cuántas caras, vértices y aristas tendrá la figura resultante. Comprueba si sigue cumpliendo el Teorema de Euler.

In a modern art exhibition, there is a display case in the shape of a cuboid with dimensions $80 \text{ cm}$ long, $60 \text{ cm}$ wide and $1 \text{ m}$ high. The artist wants to place a metal rod running through the case from a bottom corner to the opposite top corner. What must be the exact length of the rod?

The exterior of a promotional tent in the shape of a regular hexagonal pyramid needs to be painted. The side length of the hexagonal base is $4 \text{ m}$ and the apothem of the lateral face is $7 \text{ m}$. If one tin of paint covers $10 \text{ m}^2$ and the ground inside the tent is not painted, what is the minimum number of tins of paint that must be bought?

Calculate the volume of a trapezoidal prism (a water tank). The parallel sides (bases) of the trapezium measure $2 \text{ m}$ and $3 \text{ m}$, its height is $1.5 \text{ m}$, and the total length of the tank (height of the prism) is $5 \text{ m}$. If the tank is half full, how many litres of water does it contain?

Is it possible to build a regular polyhedron whose faces are regular hexagons? Justify your answer by calculating the sum of the angles of the faces that meet at a single vertex. Repeat this reasoning to explain why there are only five convex regular polyhedra.

For the local festival of a municipality in Castilla-La Mancha, the town hall is going to order a tent for craft stalls in the shape of a regular decagonal prism (a $10$-sided base). The structure is assembled using aluminium bars for the edges and spherical anchoring pieces at the vertices. Before consulting the technical catalogue, the councillor estimates off the top of his head: «For that tent, we will need about $40$ anchors and close to $50$ aluminium bars». First, make a quick mental estimate and explain whether the councillor's figures seem coherent or disproportionate to you. Then, calculate exactly how many faces ($C$), vertices ($V$), and edges ($A$) this prism has, verify that Euler's Theorem holds, and assess the error in the councillor's initial estimate.

An architecture studio in Galway designs a bioclimatic pavilion for a public park. The building is shaped like a right hexagonal prism topped at the upper part by a roof in the shape of a hexagonal pyramid, whose base coincides exactly with the top base of the prism (so that this surface remains inside and does not count as an exterior face). Model this building mathematically as a single polyhedron and answer:

1. a) How many exterior faces does it have in total and what geometric shape does each one have?
2. b) How many vertices and how many edges make up the complete exterior structure?
3. c) Check whether this composite polyhedron satisfies Euler's Theorem ($C + V = A + 2$) and explain why.

A video game design studio is creating a new "power-up" shaped like a convex polyhedron. The artists have decided that the object will have exactly $14$ faces and $24$ edges.

1. a) According to Euler's Theorem ($C + V = A + 2$), how many vertices must the 3D model have for it to be geometrically possible?

1. b) If the final polyhedron turns out to be a cuboctahedron (a polyhedron made up of squares and equilateral triangles) and we know that exactly $4$ edges meet at each of its vertices, check whether your result from part a) matches the real structure of this solid. To do so, count the edges from the number of vertices and the number of edges that meet at each one.

You are building a custom PC and want to install a latest-generation graphics card. The computer case (the tower) is a cuboid with dimensions $45\text{ cm}$ high, $20\text{ cm}$ wide and $40\text{ cm}$ long. The graphics card is very long, and you want to know if it would fit diagonally, joining a bottom corner on the front to the opposite top corner on the back. Calculate the length of the space diagonal of the cuboid and determine whether a card $62\text{ cm}$ long would fit inside.

An artisan jewellery company wants to make a silver pendant in the shape of a regular octahedron. If each edge of the octahedron measures $12\text{ mm}$, calculate the total surface area of silver needed to cover the pendant (the total area). Hint: Remember that a regular octahedron is made up of eight congruent equilateral triangles. You will first need to calculate the height of one of these triangles using Pythagoras' theorem.

An architect has designed a skylight for a shopping centre in the shape of a regular square pyramid. The base is a square with sides of $6\text{ m}$. If the area of the glass surface (the lateral surface area of the pyramid) is $60\text{ m}^2$, calculate the actual height of the pyramid from the centre of the base to the apex. Note: Be careful; you must first find the apothem of the pyramid using the lateral surface area and then apply Pythagoras' theorem to find the height.

Imagine a regular tetrahedron and a regular octahedron, both with the same edge length $a$. A scientist claims that if we join two of these polyhedra by one of their faces, the number of faces exposed on the outside of the new solid will always be the sum of the faces of both minus two.

1. a) Calculate the number of faces, vertices and edges of the solid resulting from joining two tetrahedra by a face.
2. b) Does Euler's formula still hold for this new resulting polyhedron? Justify your answer.

For a design and technology project, you need to build a speaker in the shape of a regular hexagonal prism. The perimeter of the hexagonal base is $42\text{ cm}$ and the height of the speaker is $30\text{ cm}$. For optimal sound, you must line the side walls with a special fabric that costs $15\text{ €/m}^2$. Calculate the total cost of the fabric needed to line the lateral surface of five identical speakers. (Remember to convert units from $\text{cm}^2$ to $\text{m}^2$).

A crystallographer is studying a gemstone in the shape of a frustum of a regular square pyramid. The lower base is a square with sides of $10\text{ cm}$, the upper base is a square with sides of $4\text{ cm}$, and the height of the frustum (distance between the bases) is $4\text{ cm}$.

1. a) Calculate the length of the apothem of the lateral faces (which are trapezia).
2. b) Calculate the total surface area of the gemstone.

An open-topped metal box (without a lid) in the shape of a cube and with a space diagonal measuring exactly $10\sqrt{3}\text{ cm}$ needs to be manufactured.

1. a) Calculate the edge length of this box.
2. b) If the metal used weighs $2.5\text{ grams}$ per $\text{cm}^2$, what will be the total mass of the box once manufactured? (Remember that the box has no lid).

The Science Gallery in Dublin commissions a hollow polyhedral sculpture made of $32$ faces: $12$ regular pentagons and $20$ regular hexagons. Exactly $3$ edges meet at each vertex of the polyhedron. To calculate the number of bars (edges) and connectors (vertices) that need to be ordered from the factory, two students propose different methods:

• Method A (Aoife): First calculate the edges by multiplying the number of faces by their sides and dividing by $2$ because each edge belongs to two faces: $A = (12 \times 5 + 20 \times 6) : 2$. Then, solve for the vertices using Euler's formula ($C + V = A + 2$).

• Method B (Seán): First calculate the vertices taking into account that each pentagon contributes $5$ corners and each hexagon $6$, but each vertex is shared among $3$ faces: $V = (12 \times 5 + 20 \times 6) : 3$. Then, calculate the edges by multiplying $V$ by $3$ and dividing by $2$.

Evaluate the validity of both methods and explain which one seems safer to you if you did not remember Euler's formula. Finally, calculate the exact number of faces, edges, and vertices of the sculpture and verify that both procedures lead to the same result.

At a school robotics competition in Galway, a team presents the technical specifications of a protective casing. They claim to have designed a closed convex polyhedron composed exclusively of $7$ triangular faces.

1. a) Knowing that each triangle has $3$ sides and that each edge of the polyhedron is shared by exactly two adjacent faces, calculate how many edges that supposed body should have. What mathematical problem do you find with the result obtained?
2. b) If the casing is redesigned to have $8$ triangular faces, how many edges will it have? Calculate how many vertices it must necessarily have by applying Euler's Theorem.
3. c) Generalise the reasoning: prove why it is geometrically impossible to build a closed polyhedron whose faces consist exclusively of an odd number of triangles.