Polyhedrons

We have learnt that the figures having a length only are known as the one-dimensional figure. Figures having length and breadth are known as the 2-dimensional figures. A polygon, a circle, etc are 2-d figures. Object and shapes having length breadth and height are known as 3-d shape. Let us now visualize some more solid shapes like Polyhedron.

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Polyhedron

A solid shape bounded by polygons is called a polyhedron. The word polyhedral is the plural of word polyhedron.

• Faces:  Polygons forming a polyhedron are known as its faces.
• Edges: Line segments common to intersecting faces of a polyhedron are known as its edges.
• Vertices: Points of intersection of edges of a polyhedron are known as its vertices.

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• 2D and 3D Figures
• Mapping Space

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In a polyhedron, three or more edges meet at a point to form a vertex. Following are some example of polyhedrons:

• Cuboids
  

• Cube

• Pyramid

• Triangular pyramid

Regular Polyhedron

A polyhedron is said to be a regular polyhedron if its faces are made up of regular polygons and the same number of faces meet at each vertex. This means that the faces of a regular polyhedron are congruent regular polygons and its vertices are formed by the same number of faces. A cube is a regular polyhedron but a cuboid is not a regular polyhedron as its faces are not congruent rectangles.

It is not regular because its faces are congruent triangles but the vertices are not formed by the same number of faces. Clearly, 3 faces meet at A but 4 faces meet at B.

Convex Polyhedron

If the line segment joining any two points on the surfaces of a polyhedron entirely lies inside or on the polyhedron, then it is said to be a convex polyhedron. Otherwise, it is known as the concave polyhedron.

Prisms and Pyramids

Two important members of polyhedron family are prism and pyramid. So, let us know more about these two polyhedrons.

Prism

A prism is a solid, whose side faces are parallelograms and whose bases are congruent parallel rectilinear figures.

In fig, there is a prism whose bases are rectilinear figures ABCDE and A’B’C’D’E’.

• The Base of a Prism: The end on which a prism may be supposed to stand is called the base of the prism. In fig ABCDE and A’B’C’D’E’ are the bases of the prism. Every prism has two bases.
• The Height of a Prism: The perpendicular distance between the ends of a prism is called the height of the prism. In fig B’F is the perpendicular distance between the ends ABCDE and A’B’C’D’E’. So it is the height of the prism shown in fig.
• Axis of prism: The straight line joining the centres of the ends of the prism is called the axis of the prism. In fig. a straight line passing through O’ and O is the axis of the prism.
• Length of prism: A length of a prism is a portion of the axis that lies between the parallel ends. In fig. OO’ is the length of the prism.
• Lateral faces: All faces other than the basis of a prism are known as its lateral faces. In fig., ABB’A’, BCC’B’, CDD’C’, etc are lateral faces.
• Lateral edges: The lines of intersection of the lateral faces of a prism are called the lateral edges of the prism. In fig. AA’, BB’, CC’, DD’ and EE’ are the lateral edges of the prism.

Regular Prism

A prism is called a regular prism if its ends are a regular polygon.

Right Prism

A prism is called a right prism if its lateral edges are perpendicular to its ends (bases). Otherwise, it is said to be an oblique prism.

Triangular Prism

Prism is called a triangular prism if its ends are a triangle. A right prism is called a right triangular prism if its ends are triangles. In other words, a triangular prism is called a right triangular prism if its lateral edges are perpendicular to its ends.

A prism is said to be a quadrilateral prism or a pentagonal prism or a hexagonal prism, etc according to the number of sides in the rectilinear figure forming the ends is four or five or six, etc. If the ends of the quadrilateral prism are a parallelogram, then it is also known as a paralleled. A quadrilateral prism with its ends as squares is called a rectangular solid or cuboids.

Pyramid

A pyramid is a polyhedron whose base is a polygon of any number of sides and whose other faces are triangles with a common vertex.

If all the corners of a polygon are joined to a point not lying in a plane, we get a pyramid. The figure shows a pyramid ABCDE. The base of this prism is the pentagon ABCDE and triangles VAB, VBC, VDE, and VEA are five faces.

• Vertex: The common vertex of the triangular faces of a pyramid is called the vertex 0of the pyramid. In the fig. V is the vertex of the pyramid.
• Height: The height of a pyramid is the length of the perpendicular from the vertex to the base. In the fig. VP is the height of the pyramid.
• Axis: The axis of a pyramid is a straight line joining the vertex to the central point on the base. In fig. VO is the axis of the pyramid.
• Lateral edges: The edges through the vertex of a pyramid are known as its lateral edges.
• Lateral faces: The side faces of a pyramid are known as its lateral faces.

Solved Example

Question 1: The diagram is an example of:

a. Regular polyhedron                   b. Irregular polyhedron

c. Concave polyhedron                  d. Convex polyhedron

Answer: a. Regular polyhedron. Dodecahedron is a regular polyhedron as the faces of this dodecahedron are congruent and are regular polygons.

Question 2: What are 5 polyhedrons?

Answer: Polyhedrons are platonic solid, also all the five geometric solid shapes whose faces are all identical, regular polygons meeting at the same three-dimensional angles. In addition, we known as the five regular polyhedra, they consist of the tetrahedron or pyramid, cube, octahedron, dodecahedron, and icosahedron.

Question 3: Is cone a polyhedron?

Answer: No, a cone is not considered a polyhedron although it is a solid shape. Furthermore, a cone has one circular base and it does not have nay face because faces are regular polygons, meaning they have straight sides.

Question 4: Is the empty set a polyhedron?

Answer: Basically, it depends upon your convention. If a set defined by linear inequality can certainly be empty. Whether you can call it a polyhedron or not, it certainly depends upon the convention you choose.

Question 5: Is a cube a polyhedron?

Answer:It refers to a 3D solid object that is bound by six square faces, facets or sides, among which three meet at each other at the vertex.  Furthermore, a cube is the only regular hexahedron and is the five platonic solids. Moreover, it is the only convex polyhedron whose faces are all squares.