Geometry - Polyhedron
- What is a Polyhedron?
- Definition of a Polyhedron
- Components of a Polyhedron
- Types of Polyhedra
- Special Polyhedra
- Properties Comparison of Different Polyhedra
- Euler’s Formula
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is a Polyhedron?
A polyhedron is a three-dimensional shape that is bound by polygons. These polygons are two-dimensional faces that enclose a single region that is formed by the enclosing edges. Edges meet at a point that is called a vertex. Therefore, a vertex is a point in a polyhedron where two or more edges meet. Examples of polyhedrons are a prism, pyramid, diamond, cube, octahedron, etc. These three-dimensional shapes are called polyhedrons.
Definition of a Polyhedron
In solid geometry, polyhedrons are three-dimensional objects that are made up of polygonal faces, straight lines (edges), and vertices. More than one Polyhedron is called a polyhedrons or polyhedra.
A image of polyhedra is shown below.
Components of a Polyhedron
- Faces: Faces in a polyhedron are two-dimensional polygons such as triangles, quadrilaterals, pentagons, hexagons, etc., Each face is enclosed by straight lines forming a three-dimensional polyhedron.
- Edges: Edges are straight lines where two or more faces meet.
- Vertices: Vertices are the corner points formed by three or more edges more than one vertex are called as vertices.
Types of Polyhedra
- Regular Polyhedron: A regular polyhedron is a polyhedron whose all faces are identical regular polygons i.e. all the edges are of the same length. Examples of regular polyhedra are tetrahedron, hexahedron, octahedron, etc. These are also called as platonic solids.
- Irregular polyhedron: An irregular polyhedron is a polyhedron whose faces i.e. polygons are not regular as well and all the faces are not same. So, an irregular polyhedron can have a combination of different polygons like a triangle, rectangle, or octagon.
- Convex polyhedra: A polyhedron is called a convex polyhedron if the line drawn by joining any two points on the surface of the polyhedron passes over the surface of the polyhedron or lies completely inside the polyhedron. Platonic solids are considered convex polyhedrons. The image below shows a convex polyhedron.
- Concave polyhedra: A polyhedron is a concave polyhedron if a line segment drawn by joining any two points on the surface lie outside the polyhedron. In a convex polyhedron, at least one angle is greater than `180°`. The image below shows a concave polyhedron.
Special Polyhedra
Prism: A prism consists of two regular faces (polygons) in which one is the base and the other is the top. The top and base are congruent to each other. The top and base can be triangular, rectangular, pentagon, hexagon, etc. The top and base are connected by lateral faces that are rectangles or parallelograms. The number of lateral faces count to the number of sides of the top and base. Examples of prisms are shown in the image below.
Pyramid: A pyramid consists of one base i.e. of any polygonal shape. Lines are extended from the vertices of the polygonal base and they meet at one point that is called an apex or vertex. So, the components of a pyramid are lateral faces: which are formed with sides of the base and the apex, apex: which is a single point lying above the base, and edges: which are the lines that connect the vertices of the base and the apex. Different types of pyramids are shown in the image below.
Platonic solids: Regular polyhedra are also known as platonic solids. The platonic solids consist of five three-dimensional shapes. Platonic solids are characterized by regularity in shapes, dimensions, angles, symmetry, and faces.
The key characteristics of platonic solids are as follows:
- Regular faces: The faces of platonic solids are regular polygons which means all sides and angles are equal.
- Equal angles: All angles that are formed by the intersection of the faces of platonic solids are of perfectly equal measure.
- Congruent edge lengths: All edges of a platonic pyramid are of equal length.
- Symmetry: Platonic pyramids are highly symmetrical shapes. They have rotational symmetry as well.
- Convexity: All platonic solids have the property of convexity which means that no portion of the faces or edges passes through the interior of the solids.
Examples of platonic solids are tetrahedrons,hexahedrons, and icosahedrons. Five platonic solids are shown in the image below.
Properties Comparison of Different Polyhedra
Euler’s Formula
Euler’s formula is a unique formula that gives the relationship between a number of faces, edges, and vertices of a polyhedron. Mathematically, it can be expressed as,
`V-E+F=2`
Where
`V =` Number of vertices
`E =` Number of edges
`F =` Number of faces
A given polygon can be verified with the above formula for its existence. Also, if two of the three values are given for a particular polygon, a third can be calculated. For example, if the number of faces and vertices is given, then the number of edges can be determined.
For example, if a polygon has `12` edges with `10` vertices, then its number of faces can be determined as follows
Here, `E = 12` and `V = 10`.
By Euler’s formula,
`V-E+F=2`
`10-12+F=2`
`F=2+2`
`F=4`
Therefore, the number of faces is `4`.
Solved Examples
Example `1`: A convex polyhedron is defined by `7` faces and `11` edges. How many vertices does it have?
Solution:
Here number of faces, `F=7` and number of edges `E=11`
Then by Euler’s formula
`V-E+F=2`
`V-11+7=2`
`V=4+2`
`V=6`
Therefore, the number of vertices are `6`.
Example `2`: A polyhedron has `6` edges, `8` vertices, and `7` faces. Verify whether it is a polyhedron or not.
Solution:
Here number of faces, `F=7`, number of vertices `V=8`, and number of edges `E=6`.
Then by Euler’s formula
`V-E+F=2`
`8-6+7=2`
`9 ≠2`
Therefore, the above solid is not a polyhedron.
Example `3`: How many irregular polyhedrons exist?
Solution:
The irregular polyhedron is infinite as the property of an irregular polyhedron is to have combinations of different lengths of its edges and measures of different angles.
Example `4`: Identify which of the following is not a polyhedron.
Solution:
Figure `1`: The shape is three-dimensional and made of polygons, so it is a polyhedron. The solid has equilateral triangular bases, so it is a triangular prism. It has `9` edges, `5` faces, and `6` vertices.
Figure `2`: It is not a polyhedron because it has a curved surface.
Figure `3`: The solid is formed by the polygons, so it is a polyhedron. It has `6` vertices, `12` edges, and `8` faces. In this, the four edges meet at a common point, and the faces are equilateral triangles, so it is an octahedron.
Figure `4`: The solid is formed by the polygons, so it is a polyhedron. The base is square, and the edges meet at one point called a vertex, so it is a square pyramid. It has `8` edges, `5` vertices, and `5` faces.
Practice Problems
Q`1`. Which of following solids is a polyhedron ?
- `V=13, E=16, F=10`
- `V=10, E=16, F=8`
- `V=9, E=5, F=8`
- `V=11, E=5, F=3`
Answer: b
Q`2`. Which one is a convex solid?
Answer: c
Q`3`. How many edges does a rectangular pyramid have?
- `10`
- `5`
- `8`
- `12`
Answer: c
Q`4`. Identify the platonic solids from the following.
- Cuboid
- Prism with rectangular base
- Triangular prism
- Pyramid with congruent triangular base
Answer: d
Frequently Asked Questions
Q`1`. Are the all faces of the polyhedra are same?
Answer: It is misunderstood that the faces of polyhedra are the same. The regular polyhedra have the same dimensions of faces. While irregular polyhedra have different dimensions of faces.
Q`2`. Is the cone a polyhedron?
Answer: No, the cone is not a polyhedron. The property of a polyhedron is that it has faces, vertices, and edges. But, the cone has a curved surface and therefore it is not a polyhedron. Similarly, a cylinder is not a polyhedron.
Q`3`. Are all polyhedra symmetrical?
Answer: All polyhedrons are not symmetrical, although platonic solids are symmetrical. Some regular polyhedra can be symmetrical.