Convex Polyhedrons
Convex polyhedron is a shape where if a line segment joining any two points within the surface of a polyhedron is completely inside or on the shape. A polyhedron is a 3D shape that has flat faces, straight edges, and sharp vertices. All regular polyhedron such as platonic solids is considered convex polyhedrons. Let us learn more about this interesting shape, the properties, and solve an example to understand the concept better.
| 1. | Definition of Convex Polyhedrons |
| 2. | Properties of Convex Polyhedrons |
| 3. | Types of Convex Polyhedrons |
| 4. | FAQs on Convex Polyhedrons |
Definition of Convex Polyhedrons
A convex polyhedron is just like a convex polygon. If a line segment joining any two points on the surface of a polyhedron entirely lies inside the polyhedron, it is called a convex polyhedron.
Meaning of Convex Polyhedrons
A polyhedron is considered to be convex when its surface i.e. face, edge, and vertex, does not intersect itself and a line segment joining any two points inside of a polyhedron is within the interior of the shape. Some of the real-life examples of a convex polyhedron are cubes and tetrahedrons.
Properties of Convex Polyhedron
A convex polyhedron is also known as platonic solids or convex polygons. The properties of this shape are:
- All the faces of a convex polyhedron are regular and congruent.
- Convex polyhedrons are 3D shapes with polygonal faces that are similar in form, height, angles, and edges.
- In a convex polyhedron, all the interior angles are less than 180º.
- The diagonals of the shape lie within the interior surface.
- The number of faces meets at each vertex.
Types of Convex Polyhedron
There are 5 types of convex polyhedron that we learn in geometry, they are:
- Tetrahedron - A tetrahedron is known as a triangular pyramid in geometry. The tetrahedron consists of 4 triangular faces, 6 straight edges, and 4 vertex corners. It is a platonic solid which has a three-dimensional shape with all faces as triangles
- Cube - A cube is a 3D solid object with 6 square faces and all the sides of a cube are of the same length. The cube is also known as a regular hexahedron that is a box-shaped solid with 6 identical square faces.
- Octahedron - An octahedron is a convex polyhedron with 8 faces, 12 edges, and 6 vertices and at each vertex 4 edges meet. The faces of an octahedron are shaped like an equilateral triangle.
- Dodecahedron - A dodecahedron is a convex polyhedron that consists of 12 sides and 12 pentagonal faces.
- Icosahedron - An icosahedron is a convex polyhedron with 20 faces. The icosahedron's definition is derived from the ancient Greek words Icos (eíkosi) meaning 'twenty' and hedra (hédra) meaning 'seat'.
Euler's Formula for Proof of Existence
According to Euler's formula, for any convex polyhedron, the Number of Faces plus the Number of Vertices (corner points) minus the Number of Edges always equals 2. Which is written as F + V - E = 2. Let us take apply this in one of the platonic solids - Icosahedron. Icosahedron has 20 faces, 30 edges, and 12 vertices, hence F + V - E = 20 + 12 - 30 = 2.
Related Topics
Listed below are a few topics related to a convex polyhedron, take a look.
Examples on Convex Polyhedrons
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Example 2: Find the convex polyhedron from the following figures.
Solution:
Among the given shapes, (a), (d), and (f) have all their vertices pointing outwards. Also in (a), (d), and (f), all the interior angles measure less than 180°. Therefore, these are the only convex polyhedron among the given shapes. Therefore, a), d) and f) are convex polyhedrons.
FAQs on Convex Polyhedrons
What are Convex Polyhedrons?
If the line segment joining any two points of the polyhedron is contained in the interior and within the surface of a polyhedron, then the polyhedron is said to be convex.
How Do You Identify Convex Polyhedrons?
A convex polyhedron is known as platonic solids or convex polygons. These shapes can be recognized as convex when the interior angles are less than 180° and the vertex is always pointing outwards. A convex polyhedron can also be identified when one side of the shape is bulging outwards.
What is the Difference Between Convex Polyhedron and Regular Polyhedron?
The main difference between a convex polyhedron and a regular polyhedron is the interior angles. In a convex, the interior angles are measured at less than 180° whereas in a regular polyhedron the interior angles are congruent.
What Makes a Polyhedron Convex?
A polyhedron is considered to be convex when a line segment connecting any two points on the surface of the shape lies within the interiors of the shape. These points can be any two points within the polyhedron.
What is a Regular Convex Polyhedron?
A convex polyhedron is said to be regular when the faces of the shape are congruent and the angles are also congruent. Some of the examples of the shapes are regular polyhedron, cube, regular dodecahedron, etc.