A dodecahedron is a three-dimensional figure having twelve faces that are pentagonal in shape. All the faces are flat 2-D shapes. There are five platonic solids and dodecahedron is one of them. Platonic solids are convex polyhedra in which the faces are made up of congruent regular polygons with the same number of faces meeting at each of their vertices. A dodecahedron is made up of 12 congruent pentagons with 3 pentagonal faces meeting at each of its 20 vertices. There are two types of dodecahedrons - regular and irregular dodecahedrons.
|1.||What is a Dodecahedron?|
|5.||FAQs on Dodecahedron|
What is a Dodecahedron?
Dodecahedron is derived from the Greek word "dōdeka" means "12" and "hédra" means "face or seat" that shows that it is a polyhedron with 12 sides or 12 faces. Hence, any polyhedra with 12 sides can be called a dodecahedron. It is made up of 12 pentagonal faces. The dodecahedron definition can be better understood using this illustration.
Let's assume that you are living in one of the platonic solids, and you have a house in one of the vertices. Your neighbors are living in the other vertices. Every morning you go on a jog, you run in a straight line, never turn or change your path. Is it possible to go in a straight line and return to your house without going through your neighbors' houses? You'd most likely assume that the answer is - no!
For over 2000 years, mathematicians have known about platonic solids but recently, a group of 3 mathematicians - Jayadev Athreya, David Aulicino, and Patrick Hooper found the solution to this problem in a dodecahedron. The dodecahedron has 31 such different paths.
The net of a dodecahedron helps us to identify the faces, edges, and vertices. You can make your own dodecahedron shape from the dodecahedron net shown below.
Steps to make a dodecahedron shape:
- Take the printout of the dodecahedron net.
- Fold along the inside lines.
- Glue the same colored lines to form a convex dodecahedron shape.
A regular dodecahedron has 12 regular pentagonal sides. You can see in the image of the dodecahedron net shown above that there are 12 pentagonal sides on a dodecahedron.
The surface area of a Dodecahedron ≈ 20.64 × a2 square units (where a is the length of one side)
The Volume of a Dodecahedron ≈ 7.66 × a3 cubic units (where a is the length of one side)
Properties of Dodecahedron
Let us learn some important properties (sides, edges, shapes, vertices, angles) related to the dodecahedron.
- Sides - A dodecahedron has 12 pentagonal sides.
- Edges - A dodecahedron has 30 edges.
- Vertices - It has 20 Vertices (corner points), and at each vertex 3 edges meet.
- It has 160 diagonals.
- The sum of the angles at each vertex is, 3 x 108° = 324°.
- Angles - The angle between sides of the pentagon: 108° and the angle of intersection of 2 adjacent faces (Dihedral angle): 116.56505°.
- Shapes - Dodecahedron shapes can be seen in many real-life situations like the Roman dodecahedron, dodecahedron dice, etc.
Dodecahedron Tips and Tricks:
- Tetrahedron, cube, octahedron, icosahedron, and dodecahedron are the only 5 platonic solids.
- An icosahedron has the maximum number of faces (i.e., 20).
- The dodecahedron and icosahedron have an equal number of edges, i.e., 30.
- Icosahedron is called dual of dodecahedron as both of them have the same number of edges. Icosahedron is the platonic shape that has the largest number of faces and has the largest volume of all the platonic solids.
- According to the Greek philosopher Plato, the tetrahedron symbolizes fire; the cube symbolizes the earth; similarly, all the platonic solids symbolize something. Do you know what a dodecahedron symbolizes?
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Example 1: Kate is working on a regular dodecahedron math project. Can you help her find the ratio of the volume of the dodecahedron to its surface area for the given value of side length?
We know, volume (V) = 7.66 × a3 and surface area = 20.64 × a2, where a is the side length. On dividing the volume by surface area, we get, V/A = (7.66 × a3)/(20.64 × a2). On solving, we get V/A = 0.3711 × a.
Therefore, V/A = 0.3711 × a.
Example 2: Dominic was playing a card game, and while rolling the dodecahedral dice, he wondered about the volume of that dice. Can you find out the volume of this dodecahedral dice whose side is of length 0.43 in?
We know, dodecahedron's volume (V) = 7.66 × a3 cubic units.
Given, a = 0.43 in. Substituting the value of a in the volume formula, we get, V = 7.66 × (0.43)3 cubic inches.
Therefore, the volume of the dice is 0.609023 in3.
Example 3: The sum of the length of all the edges of a regular dodecahedron is 150 in. What is the surface area of this dodecahedron?
We know that a regular dodecahedron has 30 edges. Thus, the length of each edge is 150/30 = 5 inches.
Surface area = 20.64 × a2
On substituting a = 5, we get A = 20.64 × 52 square inches.
Therefore, the surface area is 516 in2.
FAQs on Dodecahedron
What is a Regular Dodecahedron?
A regular dodecahedron is a dodecahedron with 12 pentagonal faces, all are of the same length. It is one of the 5 platonic solids. It has a total of 20 vertices, 30 edges, and 160 diagonals that includes 60 face diagonals and 100 space diagonals.
What Types of Polygons are the Faces of a Dodecahedron?
A dodecahedron has 12 pentagonal faces. Pentagons are polygons with 5 sides and 5 interior angles.
Is Dodecahedron a 2D shape?
No, a dodecahedron is not a 2-D shape. It is a 3D shape, also known as a platonic solid with twelve faces. Each of those faces is a regular pentagon.
What is a 3D Pentagon called?
A three-dimensional shape that includes pentagons is a regular dodecahedron. All the faces of a regular dodecahedron are pentagon in shape.
What is the Difference Between a Dodecagon and a Dodecahedron?
The main difference between a dodecagon and a dodecahedron is that a dodecagon is a 2D shape while a dodecahedron is a solid 3D figure. The dodecagon is made up of 12 straight edges, while a dodecahedron is formed by combining 12 pentagons on 20 vertices.
Who Invented Dodecahedron?
The dodecahedron was invented by the Pythagoreans.
How many Degrees is a Dodecahedron?
The measure of the dihedral angle of a dodecahedron is 116.565°.
Is Dodecahedron a Polyhedron?
Yes, a dodecahedron is a polyhedron because a polyhedron is a solid shape whose faces are all polygons. All the faces of the dodecahedron are pentagons, which is a regular pentagon.