Polyhedra
A 20-sided die from a board game, a shoebox, the pyramids at Giza, a classic soccer ball… They are all polyhedra: closed solids whose faces are flat polygons. Count the pieces of a classic soccer ball and a mystery shows up: how many faces, edges and corners does it have, and is there a rule that ties those numbers together? By the end of this page you will have the rule, and the tools to measure any polyhedron.
Parts of a polyhedron
Turn the solid with your finger and count each kind of part. Watch the counters at the bottom.
A cube has 6 faces, 8 vertices and 12 edges. A box with a rectangular base (a rectangular prism, or cuboid) has the same counts, and its faces come in three pairs of equal rectangles: top and bottom, front and back, left and right.
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.
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.
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.
Euler's formula
The numbers of faces, vertices and edges are not independent. For the cube: $6 + 8 = 14$ and $12 + 2 = 14$.
Why does it work? Slice a corner off a solid. You add 1 face, 2 vertices (one vertex disappears, three new ones appear) and 3 edges. The left side grows by $1 + 2 = 3$ and the right side grows by 3. The balance never breaks.
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…
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.
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.
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.
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,…
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.
Prisms and pyramids
Most polyhedra you meet are one of two families.
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Prism: two bases that are equal and parallel polygons, joined by lateral faces that are parallelograms. In a right prism the lateral faces are rectangles.
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Pyramid: one base and triangular lateral faces that meet at a single point, the apex.
Both are named after the base: triangular, square (quadrangular), pentagonal, hexagonal… A pyramid or prism is regular if it is right and its base is a regular polygon.
A pyramid has two important segments:
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the height $h$, from the apex straight down to the center of the base;
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the slant height $s$, the height of one triangular lateral face (it runs along the surface).
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.
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.
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)…
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).
Right triangles inside solids
Many measurements in a solid are hidden inside a right triangle, so the Pythagorean Theorem does the work. The first place to look is the pyramid.
Pyramids hide a second right triangle. The lateral edge $e$ (apex to a corner of the base), the height $h$ and the distance $R$ from the center of the base to a corner satisfy $e^2 = h^2 + R^2$. For a square base with edge $\ell$, $R$ is half the diagonal of the square, $R = \ell\sqrt{2}/2$.
Two more right triangles show up in bases.
An isosceles trapezoid with bases $B$ and $b$ and legs $\ell$ has two equal "overhangs" of length $\frac{B - b}{2}$ on each side. Drop a perpendicular from each top corner to make a right triangle:
$$h^2 = \ell^2 - \left(\tfrac{B - b}{2}\right)^2$$For a regular hexagon with side $s$, the apothem comes from the right triangle with hypotenuse $s$ and one leg $s/2$ (half a side): $a^2 = s^2 - (s/2)^2$. For a hexagon of side 10 this gives $a^2 = 100 - 25 = 75$, so $a = \sqrt{75} \approx 8.66$.
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…
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.
Surface area
The surface area is the total area of all faces. It has two parts:
$$\text{Surface area} = \text{lateral area} + \text{area of the bases}$$Why? If you unroll the lateral faces of a prism, you get one big rectangle with width $P$ and height $h$. The lateral faces of a pyramid are triangles with base $\ell$ and height $s$, and their areas $\tfrac{1}{2}\ell s$ add up to $\tfrac{1}{2} P s$.
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…
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.
Volume
A prism is a stack of layers, each with area $B$, piled up to height $h$. That is why $V = Bh$. A pyramid with the same base and height as a prism fills exactly one third of it: three equal pyramids fit into a cube. For example, a pyramid with a $6 \times 6$ base and height 5 has $V = \tfrac{1}{3} \cdot 36 \cdot 5 = 60$.
For capacity in gallons, use $1\ \text{ft}^3 \approx 7.48$ gallons.
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…
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 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…
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:
- a) Calculate the area of the base of the tank (you will first need to find the height of the trapezoid).
- b) Calculate the total area of the tank.
- 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…
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 $1,21 \cdot L$:
- a) Calculate the apothem of the pyramid.
- b) Calculate the amount of canvas needed to cover the lateral walls of the tent.
- c) Calculate the volume of air the tent contains.
Regular polyhedra
A polyhedron is regular if it is convex, all its faces are equal regular polygons, and the same number of faces meet at every vertex. There are exactly five, the Platonic solids:
| Polyhedron | Faces are | $F$ | $V$ | $E$ |
|---|---|---|---|---|
| Tetrahedron | equilateral triangles | 4 | 4 | 6 |
| Cube (hexahedron) | squares | 6 | 8 | 12 |
| Octahedron | equilateral triangles | 8 | 6 | 12 |
| Dodecahedron | regular pentagons | 12 | 20 | 30 |
| Icosahedron | equilateral triangles | 20 | 12 | 30 |
Check any row with Euler: for the icosahedron, $20 + 12 = 30 + 2$. ✓
Notice the pattern: the cube has $(F, V) = (6, 8)$ and the octahedron has $(8, 6)$. If you put a dot at the center of each face of a solid and connect the dots of neighboring faces, you get the dual polyhedron. Faces and vertices swap places, and the number of edges stays the same. The cube and the octahedron are duals, the dodecahedron and the icosahedron are duals, and the tetrahedron is dual to another tetrahedron.
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.
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…
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.
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…
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:
- a) What name is given to the resulting polyhedron (its dual polyhedron)?
- b) How many faces and how many vertices will this new polyhedron have? Check if it satisfies Euler's formula.
The diagonal of a box
Can a long object (a hockey stick, a javelin, a fluorescent tube) fit in a box if you lay it from one far corner to the opposite far corner? That longest straight segment is the space diagonal. Use the Pythagorean Theorem twice: first on the base, then on a vertical triangle.
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…
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 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…
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).
Truncating a polyhedron
Truncating means slicing off a corner with a flat cut. Here a cube loses one corner, using a plane through the midpoints of the three edges that meet there.
If you cut all the corners and the cuts touch at the midpoints of the edges, you cannot just add the changes one by one. Count the final solid directly: the vertices are the midpoints of the original edges, the faces are the original faces (now smaller) plus one new face per corner, and each edge of the new solid borders one of the new faces.
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…
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).
- a) How many faces, vertices and edges does the new resulting polyhedron have?
- b) Is Euler's Theorem still satisfied after the cut?
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…
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.
- a) Calculate the number of faces, vertices and edges of the resulting polyhedron.
- b) Check if the new polyhedron still satisfies Euler's Theorem.
- c) What type of polygons form the faces of the new trophy?
Similar solids
Two solids are similar if one is an enlargement of the other by a scale factor $k$. Every length (edges, heights, apothems) is multiplied by $k$. Areas are multiplied by $k^2$, and volumes by $k^3$.
Why? Area measures two dimensions, so each dimension picks up a factor $k$: $k \cdot k = k^2$. Volume measures three dimensions: $k^3$.
If you are given the volume ratio, go backward: $k = \sqrt[3]{\text{volume ratio}}$. A volume 27 times larger means $k = 3$.
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…
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:
- a) What is the scale factor $k$ between both solids?
- b) Calculate the height and total area of the second prism based on the similarity relationships for areas and volumes.
Frustums
Slice a pyramid with a plane parallel to its base. The top piece is a smaller pyramid similar to the original. The bottom piece is a frustum (a pyramid with its tip cut off).
To find the volume of a frustum, subtract the small pyramid from the full one:
$$V_{\text{frustum}} = V_{\text{full pyramid}} - V_{\text{small pyramid}}$$Similar triangles in the side view give the missing height: if the top base is $b$ and the bottom base is $B$, then (small height) ÷ (full height) $= b \div B$.
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…
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 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…
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).
- a) Calculate the base edge of the upper frustum of the pyramid.
- b) Calculate the volume of the lower frustum of the pyramid resulting from removing the upper part.
Composite solids and symmetry
Many real objects are two solids glued together or one solid with a piece carved out.
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Joined solids: add the volumes. For surface area, add all the faces, except the faces that are glued together (they are hidden).
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Carved solid: subtract the volume of the hollow. For surface area, remove the opening from the outside face and add the walls of the hollow.
A plane of symmetry cuts a solid into two halves that are mirror images of each other. Some examples:
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A box that is not a cube has 3 planes of symmetry.
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A cube has 9.
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A regular square pyramid has 4, all vertical.
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A regular triangular prism has 4: three vertical planes and one horizontal plane halfway up.
When you stack solids, only the planes that work for both parts, and for the whole shape, survive. A horizontal plane stops being a plane of symmetry if the top and the bottom of the object are different.
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…
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.
- a) Calculate the volume of granite remaining after extracting the pyramid.
- 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…
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 $288$ $cm^3$.
- a) Calculate the height of the pyramid.
- b) Calculate the maximum distance between the apex of the pyramid and any of the vertices of the bottom base of the cube.
- c) How many planes of symmetry does this composite part have?
Back to the start: a classic soccer ball is a truncated icosahedron: 12 regular pentagons and 20 regular hexagons. That is $F = 32$ faces. Each of the 60 vertices lies on 3 faces, and the 90 edges each belong to 2 faces. Check: $32 + 60 = 92 = 90 + 2$ ✓. Euler's formula works, even on a soccer ball. (The air pressure puffs the flat faces out, so the real ball is "almost" a polyhedron.)