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 rectangular prism. 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 center of the base) and the segment marked with the letter $a$ (height of a lateral face).

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 rectangular prism 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 centers 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.

A logistics company uses boxes shaped like rectangular prisms to transport sports equipment. A customer needs to send a training javelin that measures exactly $2.25$ meters 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 box, 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 trapezoid. The parallel sides (bases) of the trapezoid 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 trapezoid).
2. b) Calculate the total surface area of the tank.
3. c) If it is filled to $80\%$ of its capacity, how many liters 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 center is $10$ m. Given that the apothem of a regular octagon with side length $L$ is approximately $1.21 \cdot L$:

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 rectangular prism 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 center 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?