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?

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).

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 $6\text{ cm}$.

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.

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.

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).

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.