Polyhedra
A die, a shoebox, the pyramids of Egypt, a classic football (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
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Prism: two equal, parallel bases, and lateral faces that are parallelograms (rectangles if it is a right prism). If the bases are rectangles, it is a cuboid (a box).
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Pyramid: one base and triangular lateral faces that meet at a vertex, the apex.
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They are named after the 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 centre 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 centre 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 every 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:
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Tetrahedron
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Cube (Hexahedron)
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Octahedron
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Dodecahedron
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Icosahedron
The available polygons are: Equilateral triangle, Square and Regular pentagon.
Diagonals: Pythagoras in space
The diagonal of a cuboid with dimensions $a \times b \times c$ is found by applying Pythagoras twice (first in 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 cuboid. 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 cuboid. 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 cuboid with dimensions $40 \text{ cm} \times 30 \text{ cm} \times 120 \text{ cm}$. We want to store a fluorescent tube inside it…
We have a cardboard box shaped like a cuboid 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.
Back to the start: The classic football is "almost" a polyhedron because it is made of flat pentagons and hexagons (a truncated icosahedron), but the air pressure curves its faces so that it rolls like a sphere.