Exercises: Polyhedra

Look at the following cube and label its key elements. State how many faces, vertices and edges it has in total. Do this in your notebook.

Check whether Euler's formula holds for a triangular prism. To do so, count its elements and apply the formula:

$C + V = A + 2$

where $C$ is the number of faces, $V$ the number of vertices, and $A$ the number of edges.

Classify the following 3D shape, stating whether it is a prism or a pyramid. Give reasons for your answer by looking at its lateral faces and its bases.

What is the name of the regular polyhedron that has 4 faces that are equilateral triangles? Draw a simple sketch of its shape and state how many vertices it has.

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

Match each of the 5 Regular Polyhedra (Platonic Solids) with the polygon that forms its faces:

• Tetrahedron

• Cube (Hexahedron)

• Octahedron

• Dodecahedron

• Icosahedron

The available polygons are: Equilateral triangle, Square and Regular pentagon.

A gift box manufacturer wants to design a box in the shape of a cuboid. If the dimensions of the base are $10\text{ cm}$ and $15\text{ cm}$, and the height is $20\text{ cm}$, state how many of its faces are equal to each other.

If we have a convex polyhedron with 12 faces and 20 vertices (a dodecahedron), use Euler's formula to calculate the number of edges it has.

Look at the diagram of this square pyramid. Identify and name the segment marked with the letter $h$ (which goes from the vertex to the centre of the base) and the segment marked with the letter $a$ (height of a lateral face).

Lucas is studying the properties of a right pyramid whose base is a regular heptagon (a 7-sided polygon). In his notebook, he states the following: “Since the base has 7 sides, the pyramid has exactly 7 faces, 7 vertices, and 14 edges. If I apply Euler's Theorem, I get $7 + 7 = 14 + 2$, that is, $14 = 16$, which is impossible. Therefore, Euler's Theorem does not hold for pyramids with an odd number of sides in their bases.” Explain with reasoning the two counting errors that Lucas makes when identifying the elements of the pyramid, state the correct number of faces ($C$), vertices ($V$), and edges ($A$), and check whether Euler's formula holds.

You want to carry a rigid umbrella that measures exactly $55\text{ cm}$ long inside a cabin suitcase in the shape of a cuboid whose interior dimensions are $40\text{ cm}$ long, $30\text{ cm}$ wide, and $20\text{ cm}$ high.

1. a) Before calculating the exact value, estimate with reasoning whether the umbrella will fit along the interior space diagonal of the suitcase. To do this, calculate mentally the diagonal of the base ($40\text{ cm} \times 30\text{ cm}$) and bound between which two consecutive tens the total space diagonal must lie, knowing that $50^2 = 2500$ and $60^2 = 3600$.
2. b) Calculate the exact length of the space diagonal of the cuboid (rounding to two decimal places) and check whether your estimate was correct and whether the umbrella finally fits in the suitcase.

An architect is designing a structure for a modern art museum. The centrepiece is a convex polyhedron that has $12$ faces and $30$ edges. Due to an error in the plans, the number of vertices is not stated. Use Euler's formula to calculate how many vertices the sculpture has and explain, based on its elements, whether it could be a regular dodecahedron.

In a design and technology lesson, a group of students is building a box shaped like a regular hexagonal prism to store 3D printer filament. The base edge measures $10 \text{ cm}$ and the height of the prism is $25 \text{ cm}$. Calculate the total area of cardboard needed to make the box (including the lid).

Note: Remember to calculate the apothem of the base first using Pythagoras' theorem.

A perfume company has designed a bottle shaped like a regular square pyramid. The base edge measures $6 \text{ cm}$ and the slant edge (connecting the apex to a vertex of the base) measures $10 \text{ cm}$. Calculate the height of the pyramid to find out if it fits on a shelf that is $9 \text{ cm}$ high.

We have a cardboard box shaped like a cuboid with dimensions $40 \text{ cm} \times 30 \text{ cm} \times 120 \text{ cm}$. We want to store a fluorescent tube inside it that is $128 \text{ cm}$ long. Is it possible to store it in a straight line joining two opposite vertices of the box? Justify your answer by calculating the space diagonal of the polyhedron.

A dice collector has a very special die shaped like a regular octahedron (8 triangular faces) where each edge measures $4 \text{ cm}$. If he decides to make a new die by joining the centres of the faces of the original octahedron:

1. a) What is the resulting polyhedron called (its dual polyhedron)?
2. b) How many faces and how many vertices will this new polyhedron have? Check whether it satisfies Euler's formula.

A trophy is shaped like a frustum of a regular square pyramid. The edges of the bottom base measure $10 \text{ cm}$, those of the top base measure $4 \text{ cm}$ and the height of the frustum is $12 \text{ cm}$. Calculate the volume of metal needed to cast this trophy.

Hint: You can use the formula for the volume of a frustum or consider the frustum as the difference between two pyramids.

A mechanical part has been designed consisting of a cube with an edge of $8 \text{ cm}$ on which a plane cut has been made that passes through the midpoints of the three edges that meet at a vertex $V$. In this way a small triangular pyramid is removed (truncation process). In the figure, the dashed red lines mark the removed pyramid and the dashed gray lines are hidden edges.

1. a) How many faces, vertices and edges does the resulting new polyhedron have?

1. b) Does Euler's theorem still hold after the cut?

Calculate the volume of a prism whose base is an isosceles trapezium. The parallel sides (bases) of the trapezium measure $12 \text{ cm}$ and $8 \text{ cm}$, and its non-parallel sides measure $5 \text{ cm}$. The height of the prism is $20 \text{ cm}$.

Note: You must first find the height of the trapezium base to calculate its area.

To calculate the length of the taut cable connecting the two furthest opposite vertices of an industrial warehouse shaped like a rectangular cuboid with dimensions $12\text{ m}$ long, $9\text{ m}$ wide, and $8\text{ m}$ high, two technicians propose different approaches:

• Method A: Apply the Pythagorean Theorem twice consecutively in two dimensions (first on the floor plane to find the base diagonal, and with that result, in the vertical right triangle containing the height).

• Method B: Directly apply the space diagonal formula for a cuboid $D = \sqrt{a^2 + b^2 + c^2}$.

Calculate the length of the cable using both methods, showing the numerical steps. Next, compare the two procedures: which one provides a better understanding of the geometric origin of the formula, and which one is more efficient in practice? Furthermore, justify whether Method B could be applied to any right prism or only to rectangular cuboids.

A craft shop in Kilkenny needs to send by post a decorative replica of a sword that is $42\text{ cm}$ long and cannot be bent or disassembled. For shipping, they choose a commercial box shaped like a rectangular cuboid with interior dimensions of $32\text{ cm}$ long, $24\text{ cm}$ wide, and $15\text{ cm}$ high.

1. a) Model the box as a geometric cuboid and calculate the maximum theoretical length that a thin straight object can have to fit inside without deforming.
2. b) Determine whether the $42\text{ cm}$ sword fits in the box and calculate how many centimetres of clearance are left over or missing relative to the space diagonal.
3. c) If the sword were $45\text{ cm}$ long, by how many centimetres at least would the height of the box need to be increased exclusively (keeping the $32\text{ cm}$ length and $24\text{ cm}$ width fixed) so that it fits exactly?

A logistics company uses cuboid-shaped boxes to transport sports equipment. A customer needs to send a training javelin that measures exactly $2.25$ metres long. The warehouse has boxes with dimensions of $200$ cm long, $80$ cm wide and $60$ cm high. By calculating the space diagonal of the cuboid, determine with reasons whether the javelin can fit inside the box without bending (assuming its thickness is negligible).

A trophy designer for an esports competition decides to create a trophy from a glass cube with edge length $12$ cm. The process involves "truncating" the cube by cutting off each of its $8$ vertices using planes that pass through the midpoints of the three edges meeting at that vertex.

1. a) Calculate the number of faces, vertices and edges of the resulting polyhedron.
2. b) Check whether the new polyhedron still satisfies Euler's formula.
3. c) What types of polygons make up the faces of the new trophy?

A regular square pyramid has a height of $24$ cm and its base area is $100$ $cm^2$. A cut is made with a plane parallel to the base at a height of $12$ cm from the vertex (that is, through the midpoint of the height).

1. a) Calculate the base edge of the upper truncated pyramid (smaller pyramid).
2. b) Calculate the volume of the resulting lower frustum of the pyramid after removing the upper part.

We have two right prisms with regular hexagonal bases that are similar. The first has a height of $10$ cm and a total surface area of $350$ $cm^2$. If the second prism has a volume that is exactly $8$ times the volume of the first:

1. a) What is the scale factor $k$ between the two solids?
2. b) Calculate the height and total surface area of the second prism using the scale factor relationships for areas and volumes.

A water tank is shaped like a right prism whose base is an isosceles trapezium. The parallel sides (bases) of the trapezium measure $4$ m and $10$ m, and the non-parallel sides measure $5$ m each. If the height of the tank (the length of the prism) is $15$ m:

1. a) Calculate the base area of the tank (you will first need to find the height of the trapezium).
2. b) Calculate the total surface area of the tank.
3. c) If it is filled to $80\%$ of its capacity, how many litres of water will it contain?

A circus tent is to be built in the shape of a regular octagonal pyramid. The base edge measures $6$ m and the height of the tent at the centre is $10$ m. Given that the apothem of a regular octagon with side length $L$ is approximately $1.21 \times L$:

1. a) Calculate the slant height (apothem) of the pyramid.
2. b) Calculate the area of canvas needed to cover the side walls of the tent.
3. c) Calculate the volume of air inside the tent.

A granite block is shaped like a cuboid with dimensions $40 \times 40 \times 100$ cm. A sculptor wants to carve out a hollow inside it in the shape of a square pyramid whose base coincides with one of the $40 \times 40$ cm faces and whose apex reaches exactly the centre of the opposite face.

1. a) Calculate the volume of granite remaining after removing the pyramid.
2. b) Calculate the total surface area of the resulting solid (including the interior walls of the pyramidal hollow).

A mechanical component is made by joining a cube with edge length $a = 6$ cm and a regular pyramid whose base coincides with one of the faces of the cube. The total volume of the component is $288$ $cm^3$.

1. a) Calculate the height of the pyramid.
2. b) Calculate the maximum distance between the apex of the pyramid and any vertex of the bottom base of the cube.
3. c) How many planes of symmetry does this composite shape have?

We want to identify a mysterious convex polyhedron of which we only know two facts: it has exactly $30$ edges ($E = 30$) and the number of its vertices exceeds the number of its faces by $8$ ($V = F + 8$).

1. a) Set up an equation using Euler's formula ($F + V = E + 2$) and calculate precisely how many faces and how many vertices this polyhedron has.
2. b) It is known that all its faces are identical regular polygons, each with $n$ sides. Knowing that each edge is shared by exactly two faces (that is, $n \times F = 2 \times E$), deduce algebraically what type of regular polygon forms the faces of this polyhedron.
3. c) Based on the previous results, identify which of the five existing regular polyhedra it is, and calculate how many edges meet at each of its vertices using the relation $k \times V = 2 \times E$, where $k$ is the number of edges per vertex.

The urban planning department of a city council is designing a fair marquee in the shape of a right pyramid with a regular octagonal base. Each side of the octagon measures $3\text{ m}$ and the apothem of the base measures $3.62\text{ m}$. According to the technical specifications:

1. The height ($h$) of the apex must strictly be between $3.5\text{ m}$ and $5.5\text{ m}$.

2. To ensure rainwater runoff, the slant height of the pyramid ($s$) must measure at least $5\text{ m}$.

1. a) If the designers initially set a height of $h = 4\text{ m}$, model the corresponding section using a right triangle, calculate the slant height of the pyramid ($s$), and justify whether the rainwater runoff criterion is met.
2. b) If the design is modified so that the slant height of the pyramid is exactly $s = 5.2\text{ m}$, calculate the new height $h$ of the marquee and check whether it remains within the range permitted by the municipal specifications.
3. c) Determine the total number of faces, vertices, and edges of this marquee, and formally prove that it satisfies Euler's characteristic for convex polyhedra.