Polyhedra
A die, a shoebox, the pyramids of Egypt, a classic soccer ball (almost)… These are polyhedra: closed solids whose faces are polygons. Role-playing dice with 4, 8, 12, and 20 faces are also very special polyhedra: the regular ones.
Elements of a polyhedron
Rotate the polyhedron with your finger and count its elements. Look at the last line:
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.
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…
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.
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.
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.
Prisms and pyramids
-
Prism: two equal, parallel bases, and lateral faces that are parallelograms (rectangles if it is right). If it is right and the bases are rectangles, it is a rectangular prism (a box; in some books, a cuboid).
-
Pyramid: one base and triangular lateral faces that meet at one vertex, the apex.
-
They are named after their base: triangular, square, pentagonal… They are regular if the base is a regular polygon and they are right.
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.
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.
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…
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).
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…
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.
The regular polyhedra
A polyhedron is regular if all its faces are equal regular polygons and the same number of faces meet at each vertex. There are only five, the Platonic solids:
| Polyhedron | Faces | $C$ | $V$ | $A$ |
|---|---|---|---|---|
| Tetrahedron | triangles | 4 | 4 | 6 |
| Cube (hexahedron) | squares | 6 | 8 | 12 |
| Octahedron | triangles | 8 | 6 | 12 |
| Dodecahedron | pentagons | 12 | 20 | 30 |
| Icosahedron | triangles | 20 | 12 | 30 |
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…
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…
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.
Diagonals: the Pythagorean theorem in space
The diagonal of a rectangular prism with dimensions $a \times b \times c$ is found by applying the Pythagorean theorem twice (first on the base, then in the vertical triangle):
$$D = \sqrt{a^2 + b^2 + c^2}$$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…
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.
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…
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.