Before getting to the formula, let us see the history of the name ”Platonic solids”. The ancient Greeks studied the Platonic solids pretty extensively. For the namesake, the platonic solids occur in the philosophy of Plato. Plato wrote about them in his book Timaeus c. 360 B.C. where he associated the four elements of Earth (earth, air, water and fire) with the regular solid. Cude was associated with The Earth, icosahedron was clubbed for water, and tetrahedron for fire.
The five platonic solids are:
- Tetrahedron
- Cube
- Octahedron
- Dodecahedron
- Icosahedron
They all are convex regular polyhedra. These five platonic solids have different formulas.
FORMULA OF TETRAHEDRON
Tetrahedron: A tetrahedron has 4 faces, 4 vertices, 6 edges and 3 concurrent edges at a vertex:
Cube: The cube is a solid which has has 6 faces, 8 vertices, 12 edges and 3 concurrent edges at a vertex:
Octahedron: A Solid which has 8 faces, 6 vertices, 12 edges and 4 concurrent edges at a vertex.
Dodecahedron: A solid which has 12 faces, 20 vertices, 30 edges and 3 concurrent edges at a vertex.
Icosahedron: A solid which has 20 faces, 12 vertices, 30 edges and 5 concurrent edges at a vertex.
At-Home STEM Activities: Platonic Solids and Euler’s Formula
We live in a three-dimensional space. That means that solid objects around us have length, width, and depth. In geometry, we can talk about specific types of solid objects, one type being Platonic solids.
A Platonic solid is 3-D shape where each face is a regular polygon and the same number of polygons meet at each vertex. For example, a cube is a Platonic solid—all of a cube’s faces are the same size squares and three squares meet at each vertex.
In total, there are only 5 Platonic solids:
These 5 shapes have been studied since ancient times—the name “Platonic” comes from the ancient Greek philosopher Plato who used the solids to represent the five classical elements: fire, earth, air, water, and the universe.
We can follow in the footsteps of our mathematical predecessors by exploring Platonic solids—let’s start by constructing these shapes!
Building Platonic Solids
Materials:
- Straws
- Scissors
- Pipe cleaners
- Ruler (to measure straws to cut in half, thirds, and quarters)
Instructions:
Tetrahedron:
1. Cut 3 straws in half, for a total of 6 pieces of straw. Put 3 pieces of straw on pipe cleaner and twist together to form a triangle.
When your pipe cleaner gets too short, twist another on the end. Put 2 pieces of straw on the pipe cleaner. Twist together with the first triangle to form the shape below.
Attach another pipe cleaner to the corner of one triangle (not one of the corners shared by both triangles). Put a straw piece on the pipe cleaner and attach to the remaining corner to form the tetrahedron.
Cube:
1. Cut 6 straws in half, for a total of 12 pieces of straw. Put 4 pieces of straw on a pipe cleaner and twist together to form a square.
Attach another pipe cleaner and place 5 pieces of straw on it. Twist the last 4 straws you place on the pipe cleaner into a square to form the shape below.
Attach 3 pieces of pipe cleaner to each corner of one of the squares and put a piece of straw on each. Twist the pipe cleaners to the corresponding corners on the other square to form the cube.
Octahedron:
1. Cut 4 straws into thirds, for a total of 12 pieces of straws. Put 3 pieces of straw onto a pipe cleaner and twist into a triangle.
Attach another pipe cleaner and put 2 pieces of straw on in. Bend into a triangle, as shown below. Repeat this twice to make two more triangles of straws. At the end of this step, your figure should look like the last photo below.
Place one piece of straw on the pipe cleaner and attach to the first triangle you made to form the below figure.
Place one straw on the pipe cleaner and attach to the free corner of the triangle you formed in step 3. Place the last piece of straw on the pipe cleaner and attach to the open corner to complete the octahedron.
Dodecahedron:
1. Cut straws into quarters for a total of 30 pieces of straw. Put 5 pieces of straw on a pipe cleaner and bend into a pentagon.
Put 4 pieces of straw on the pipe cleaner and bend into another pentagon, as shown in the first photo below. Continue adding straws to the pipe cleaner and bending into pentagons along each side of the pentagon you formed in step 1. At the end of this step, you should have a figure that looks like the last photo below.
Attach a pipe cleaner to an unshared corner of one of the pentagons. Put 3 pieces of straw on the pipe cleaner and attach to an adjacent pentagon, as shown below.
Form three more pentagon on the three open sides of the pentagon you formed in step 3.
Complete your dodecahedron by placing a pipe cleaner and straw on the last open corners, forming the last two pentagons.
Icosahedron:
1. Cut straws into quarters for a total of 30 pieces of straw. Put 3 pieces of straw on a pipe cleaner and bend into a triangle.
Put 2 pieces of straw on the pipe cleaner and form a second triangle. Continue to form triangles, as shown below, until you have 4 triangles. Put one piece of straw on the pipe cleaner and attach to the first triangle to form a closed circle of triangles.
Form a triangle off the open side of each triangle formed in steps 1 and 2.
Connect the open corner of each triangle formed in step 3 to the adjacent triangles.
Attach a pipe cleaner to one of the open corners. Place 2 straws on the pipe cleaner and bend into a triangle. To complete your icosahedron, continue forming triangles until you have 5 triangles that meet at one point, like you formed in step 2.
You’ve built your set of Platonic solids!
Now that we have our set of Platonic solids, let’s take a closer look at them.
We built our tetrahedron using 6 straws. That means that a tetrahedron has 6 edges. Counting the faces and vertices, we see that a tetrahedron has 4 faces and 4 vertices.
How many faces, edges, and vertices do the other Platonic solids have?
Count these properties using the solids you built, then scroll down to see the answer!
Now, evaluate the following for each Platonic solid:
(number of faces) – (number of edges) + (number of vertices)
Scroll down once you’ve done this.
F – E + V =2 for all Platonic solids
This is known as Euler’s Formula, named after Swiss mathematician Leonhard Euler (pronounced like “oiler”). Euler’s Formula is a way to categorized different types of polyhedrons, and, this formula holds true for almost all polyhedrons. The only ones that it doesn’t work for, have holes in them.
For example, Euler’s Formula holds true for an octagonal pyramid (a pyramid with an octagon as a base)—the pyramid has 9 faces, 16 edges, and 9 vertices; 9 – 16 + 9 = 2—but it does not hold for a toroidal polyhedron (a polyhedron with one hole)—this shape has 24 faces, 48 edges, and 24 vertices; 24 – 48 + 24 = 0.
A polyhedron without any holes is called simple, while one with holes is called non-simple. So, by Euler’s Formula, calculating F – E + V for a simple polyhedron equals 2 and for a non-simple polyhedron doesn’t equal 2. As an extra activity, look around your home for more examples of polyhedron and calculate Euler’s Formula, classifying these objects as simple or non-simple.
If you are interested to learn more about Euler’s Formula, this article gives a good explanation, including a proof for the formula. The proof may be a little advanced, but we encourage older learners who are up for a challenge to take a look!
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