Exercises: Polyhedra

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

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

$C + V = A + 2$

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

Classify the following geometric solid, stating whether it is a prism or a pyramid. Justify your answer by looking at its lateral faces and its bases.

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

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 theorem to calculate the number of edges it has.

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

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

In the technology workshop, a group of students is building a box in the shape of a regular hexagonal prism to store 3D printer filaments. 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).

Fact: Remember to first calculate the apothem of the base using the Pythagorean theorem.

A perfume company has designed a bottle in the shape of a regular quadrangular pyramid. The base edge measures $6 \text{ cm}$ and the lateral edge (the one joining the apex to a corner of the base) measures $10 \text{ cm}$. Calculate the height of the pyramid to find out if it fits on a shelf $9 \text{ cm}$ high.

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

A dice collector has a very special one in the shape of a regular octahedron (8 triangular faces) where each edge measures $4 \text{ cm}$. If they decide to make a new die by joining the centers of the faces of the original octahedron:

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

A trophy is in the shape of a regular quadrangular frustum of a pyramid. The edges of the bottom base measure $10 \text{ cm}$, those of the top base $4 \text{ cm}$ 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 of edge $8 \text{ cm}$ to which a planar cut has been made passing through the midpoints of three concurrent edges at a vertex, thus removing a small triangular pyramid (truncation process).

1. a) How many faces, vertices and edges does the new resulting polyhedron have?
2. b) Is Euler's Theorem still satisfied after the cut?

Calculate the volume of a prism whose base is an isosceles trapezoid. The parallel sides of the trapezoid measure $12 \text{ cm}$ and $8 \text{ cm}$, and its slanted sides measure $5 \text{ cm}$. The height of the prism is $20 \text{ cm}$.

Hint: You will first need to find the height of the base trapezoid in order to calculate its area.

A logistics company uses cuboid-shaped boxes to transport sports equipment. A customer needs to send a training javelin that is exactly $2,25$ meters long. The warehouse has boxes with dimensions $200$ cm long, $80$ cm wide and $60$ cm high. Determine with reasoning, by calculating the diagonal of the cuboid, whether the javelin can travel inside the box without bending (assuming its thickness is negligible).

A trophy designer for a video game competition has decided to create a piece from a glass cube with an edge of $12$ in. The process consists of "truncating" the cube by cutting each of its $8$ vertices using planes that pass through the midpoints of the three edges that meet at that vertex.

1. a) Calculate the number of faces, vertices and edges of the resulting polyhedron.
2. b) Check if the new polyhedron still satisfies Euler's Theorem.
3. c) What type of polygons form the faces of the new trophy?

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

1. a) Calculate the base edge of the upper frustum of the pyramid.
2. b) Calculate the volume of the lower frustum of the pyramid resulting from removing the upper part.

We have two similar right prisms with regular hexagonal bases. The first has a height of $10$ in and its total area is $350$ $cm^2$. 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 both solids?
2. b) Calculate the height and total area of the second prism based on the similarity relationships for areas and volumes.

A water tank is in the shape of a right prism whose base is an isosceles trapezoid. The bases of the trapezoid measure $4$ ft and $10$ ft, and the equal sides measure $5$ ft. If the height of the tank (the length of the prism) is $15$ ft:

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

A circus tent is to be built in the shape of a regular octagonal pyramid. The base edge measures $6$ ft and the height of the tent in the center is $10$ ft. Knowing that the apothem of a regular octagon of side $L$ is approximately $1,21 \cdot L$:

1. a) Calculate the apothem of the pyramid.
2. b) Calculate the amount of canvas needed to cover the lateral walls of the tent.
3. c) Calculate the volume of air the tent contains.

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

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

A mechanical part is built by joining a cube of edge $a = 6$ in and a regular pyramid whose base coincides with one of the faces of the cube. It is known that the total volume of the part 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 of the vertices of the bottom base of the cube.
3. c) How many planes of symmetry does this composite part have?