Polyhedra
A tabletop role-playing set comes with dice that have 4, 6, 8, 12 and 20 faces. A shoe box, a pyramid at a theme park and a Rubik's Cube are also closed solids whose faces are flat polygons. These solids are called polyhedra. By the end of this page you will know why that dice set has exactly these five shapes and no others.
Faces, edges and vertices
Rotate the cube and count its parts. Watch the last line.
A polyhedron is convex if the segment between any two of its points stays inside the solid. Think of a cube, not of a "dented" shape. In this page every polyhedron is convex unless we say otherwise.
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.
Euler's formula
Count the parts of a few solids:
| Solid | $F$ | $V$ | $E$ | $F + V$ | $E + 2$ |
|---|---|---|---|---|---|
| Cube | 6 | 8 | 12 | 14 | 14 |
| Triangular prism | 5 | 6 | 9 | 11 | 11 |
| Square pyramid | 5 | 5 | 8 | 10 | 10 |
The last two columns always agree.
Some problems write this as $C + V = A + 2$, with $C$ for faces and $A$ for edges. It is the same formula.
Why does the formula work? A convex polyhedron is like a balloon. If you inflate it into a sphere, you get a map of faces, edges and vertices on the sphere. Every sphere-like map obeys $F - E + V = 2$. That is why the formula holds for every convex polyhedron.
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
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A prism has two congruent, parallel bases. Its lateral faces are parallelograms, and rectangles if the prism is right. If the bases are rectangles, it is a rectangular prism, or box.
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A pyramid has one base. Its lateral faces are triangles that meet at one point, the apex.
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Both are named by their base: triangular prism, pentagonal pyramid, and so on.
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A pyramid is regular if it is right and its base is a regular polygon. A right prism has lateral edges perpendicular to the bases.
If the base has $n$ sides, you can predict the counts:
| Solid | $F$ | $V$ | $E$ |
|---|---|---|---|
| $n$-gonal prism | $n + 2$ | $2n$ | $3n$ |
| $n$-gonal pyramid | $n + 1$ | $n + 1$ | $2n$ |
Check the pentagonal prism: $7 + 10 = 15 + 2$. ✓
A rectangular prism with dimensions $\ell \times w \times h$ has three pairs of equal faces: $\ell \times w$ (top and bottom), $\ell \times h$ (front and back) and $w \times h$ (left and right).
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.
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.
Pyramids: right triangles, area and volume
A right pyramid hides several right triangles. Learn to find them.
A word about labels. Books and problems do not all use the same letters. In some drawings the slant height (the height of a lateral face, from the apex to the midpoint of a base edge) is labeled $a$, and the height $h$ is the segment from the apex to the center of the base. Here we write $\ell$ for the slant height and keep $a$ for the apothem of the base. Always check which segment a letter stands for by looking at where it starts and ends.
The apothem of the base is the shortest distance from the center to a side. For a regular polygon, the base area is $B = \tfrac12 aP$, where $P$ is the perimeter.
Other regular bases work the same way. You only need the apothem of the base. For a regular octagon with side $s$, the apothem is about $1.21\,s$. (It comes from the right triangle with a $22.5°$ angle: $a = \dfrac{s}{2\tan 22.5°} \approx 1.207\,s$.)
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).
A perfume company has designed a bottle in the shape of a regular square pyramid. The base edge measures $6 \text{ cm}$ and the lateral edge (the one joining the apex to…
A perfume company has designed a bottle in the shape of a regular square 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.
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.
Prisms: surface area and volume
Often the first job is to find the base area. Two helpers appear again and again.
A trapezoid base. The area is $\tfrac12(b_1 + b_2)h_t$. In an isosceles trapezoid, drop two perpendiculars. Each makes a right triangle with a leg of $\tfrac{b_1 - b_2}{2}$ and the slanted side as the hypotenuse. That gives the trapezoid's height $h_t$.
A regular hexagon base. Six equilateral triangles fit around the center. The apothem $a$ is the height of one of them.
Why $a = \tfrac{\sqrt{3}}{2}s$? Split an equilateral triangle in half. You get a 30-60-90 triangle with hypotenuse $s$ and short leg $\tfrac{s}{2}$. By the Pythagorean Theorem, $a = \sqrt{s^2 - \tfrac{s^2}{4}} = \tfrac{\sqrt{3}}{2}s$.
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.
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?
The regular polyhedra
A polyhedron is regular if all its faces are congruent regular polygons and the same number of faces meet at every vertex. There are only five, the Platonic solids.
| Polyhedron | Faces | $F$ | $V$ | $E$ |
|---|---|---|---|---|
| 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 |
Duals. Mark the center of every face of a polyhedron and join the centers of faces that share an edge. You get a new polyhedron, the dual. Faces and vertices trade places: if the original has $F$ faces and $V$ vertices, the dual has $V$ faces and $F$ vertices, and the same number of edges. The cube and the octahedron are duals, the dodecahedron and the icosahedron are duals, and the tetrahedron is its own dual.
Back to the dice. The 20-sided die is an icosahedron: 20 triangles, 12 vertices, 30 edges, and $20 + 12 = 32 = 30 + 2$. In a Platonic solid every face is identical and every vertex looks the same, so no face is favored when the die lands. That is why the dice set uses exactly the tetrahedron, cube, octahedron, dodecahedron and icosahedron. A classic soccer ball is not regular. It is a truncated icosahedron with 12 pentagons and 20 hexagons: $F = 32$, $V = 60$, $E = 90$, and $32 + 60 = 90 + 2$. ✓
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 die 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 die 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.
Cutting off corners
Slice off a corner of a polyhedron with a plane. This is called truncation. Count what changes.
Faces change too. A face that loses a corner gains a side. A triangle with a corner cut off becomes a quadrilateral.
What if you cut every vertex through the midpoints of its edges? Neighboring cuts now meet at the same midpoint, so the one-corner rule no longer applies. Count the pieces of the new solid directly.
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?
The Pythagorean Theorem in space
A space diagonal of a box joins two opposite vertices through the inside. To find its length, use the Pythagorean Theorem twice: first on the base, then on a vertical triangle.
We have a cardboard box in the shape of a rectangular prism with dimensions $40 \text{ cm} \times 30 \text{ cm} \times 120 \text{ cm}$. We want to store a fluorescent…
We have a cardboard box in the shape of a rectangular prism 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 space diagonal of the polyhedron.
A logistics company uses rectangular prism-shaped boxes to transport sports equipment. A customer needs to send a training javelin that is exactly $2,25$ feet long. The…
A logistics company uses rectangular prism-shaped boxes to transport sports equipment. A customer needs to send a training javelin that is exactly $2,25$ feet long. The warehouse has boxes with dimensions $200$ in long, $80$ in wide and $60$ in high. Determine with reasoning, by calculating the diagonal of the rectangular prism, whether the javelin can travel inside the box without bending (assuming its thickness is negligible).
Frustums and similar solids
Cut a pyramid with a plane parallel to its base. The small pyramid on top is a smaller copy of the whole pyramid. The part left at the bottom is a frustum.
Two solids are similar if one is a scaled copy of the other with scale factor $k$.
To get the volume of a frustum, subtract: frustum = whole pyramid minus small pyramid.
When a problem gives the base area and tells you where the cut is, find $k$ from the heights. Then use $k$ for lengths and $k^3$ for volumes.
A trophy is in the shape of a frustum of a regular square pyramid. The edges of the bottom base measure $10 \text{ cm}$, those of the top base $4 \text{ cm}$ and the…
A trophy is in the shape of a frustum of a regular square 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 square 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…
A regular square 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.
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.
Composite solids and symmetry
A composite solid is built by joining or removing simpler solids. Break it into pieces you know. Add volumes when you join, subtract when you carve out. For surface area, think about which faces are visible and which are hidden or newly exposed.
Symmetry. A plane of symmetry cuts a solid into two mirror-image halves. A right prism with a rectangular base (all three dimensions different) has 3 planes of symmetry. A cube has 9.
A granite block is in the shape of a rectangular prism with dimensions $40 \times 40 \times 100$ in. A sculptor wants to carve a hollow inside it in the shape of a…
A granite block is in the shape of a rectangular prism with dimensions $40 \times 40 \times 100$ in. A sculptor wants to carve a hollow inside it in the shape of a square 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?