Polyhedra

Look around you in the classroom or in your bedroom and name:

1. a) An object that has the shape of a rectangular prism.
2. b) How many faces does that object have?
3. c) Are all its faces the same or are there different faces? Describe what they are like.

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

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

A video game design studio is creating a new "power-up" in the shape of a convex polyhedron. The artists have decided that the object will have exactly $14$ faces and $24$ edges. According to Euler's Theorem, how many vertices must the 3D model have for it to be geometrically possible? If the final polyhedron turns out to be a cuboctahedron (a polyhedron made of squares and equilateral triangles), and we know that exactly $4$ edges meet at each of its vertices, check if your calculations match the actual structure of this solid.

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

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

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

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

The dice used in role-playing games usually have the shapes of regular polyhedra. If you are playing with a die in the shape of an octahedron:

1. a) What shape are its faces?
2. b) How many faces does this die have?
3. c) How many edges meet at each of its vertices?

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

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

2. The regular tetrahedron is the polyhedron with the fewest possible 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.

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

A sculptor wants to create a piece by carving a wooden cube with an edge length of $10 \text{ cm}$. If they remove the four top corners by cutting through the midpoints of the edges that meet at each vertex, carefully count how many faces, vertices and edges the resulting shape will have. Check if it still satisfies Euler's Theorem.

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 to 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 Theorem still hold true for this new resulting polyhedron? Justify your answer.

Ashley wants to make a cardboard gift box in the shape of a cube. To do this, she draws different templates in her notebook. State, giving reasons, which of the following two nets allows a perfect cube to be formed and which does not.

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

For a technology project, you have 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 cover the side walls with a special fabric that costs $15\text{ \$/m}^2$. Calculate the total cost of the fabric needed to cover the lateral surface of five identical speakers. (Remember to convert units from $\text{cm}^2$ to $\text{m}^2$).

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

An artisan jewelry 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 they will need to cover the pendant (the total area). Hint: Remember that a regular octahedron is made up of eight identical equilateral triangles. You will first need to calculate the height of one of these triangles using the Pythagorean theorem.

We want to paint the outside of an advertising tent in the shape of a regular hexagonal pyramid. The side of the base hexagon measures $4 \text{ m}$ and the slant height of the lateral face measures $7 \text{ m}$. If a can of paint covers $10 \text{ m}^2$ and the floor of the tent is not painted, what is the minimum number of cans of paint that must be bought?

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

A crystallographer is studying a gemstone in the shape of a frustum of a regular square-based pyramid. The bottom base is a square with a side length of $10\text{ cm}$, the top base is a square with a side length of $4\text{ cm}$, and the height of the frustum (distance between bases) is $4\text{ cm}$.

1. a) Calculate the slant height of the lateral faces (which are trapezoids).
2. b) Calculate the total area of the gemstone.

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 that goes through the display case from a bottom vertex to the opposite top vertex. What must the exact length of the rod be?

You are building a custom PC and want to install a next-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 by placing it diagonally, joining a bottom front vertex with the opposite top back vertex. Calculate the length of the diagonal of the cuboid and determine if a $62\text{ cm}$ long card would fit inside.

We want to manufacture a metal box without a lid in the shape of a cube whose diagonal measures exactly $10\sqrt{3}\text{ cm}$.

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 top lid).

Calculate the volume of a regular square-based pyramid whose base edge measures $10 \text{ cm}$ and whose lateral edge measures $13 \text{ cm}$. To solve this, you must first calculate the slant height of the pyramid and then its height using the Pythagorean theorem.

Calculate the volume of a trapezoidal prism (a water tank). The parallel sides of the trapezoid measure $2 \text{ ft}$ and $3 \text{ ft}$, its height is $1,5 \text{ ft}$ and the total length of the tank (height of the prism) is $5 \text{ ft}$. If the tank is half full, how many gallons of water does it contain?