Question

What are regular polyhedrons? Give a few examples. Can we have a general formula for finding their surface areas?

Answer 1

Hint:The polyhedron shapes are the 3D shapes whose face comprises a polygon shape. If we are saying any polygon is a regular polygon then its all sides should be equal and all the interior angles should also be equal.

Complete answer:
In the above question, first we have to find what are regular polyhedrons and some of its examples.So, the basic definition of a regular polyhedron is: a regular polyhedron is a $3 - D$ figure whose all surfaces are regular polygons. A regular polygon is one whose all sides and interior angles are equal.

There are few examples of regular polyhedra: Two such well known polyhedra are tetrahedra (formed by four equal equilateral triangles) and cube (formed by six equal squares). Some other polygons include Octahedron formed by eight equal equilateral triangles. Similarly, there are some other polyhedra such as dodecahedron and icosahedron formed by twelve and twenty equilateral triangles respectively.

Formulae for finding the total surface areas, volumes and lateral surface areas are on the basis of the length of their sides in case of regular polygon and radius of circumscribing or inscribing circle. So, there is no general formula for finding the surface area of a regular polyhedron.

Note:There are some properties of a regular polyhedron as:
-The vertices of a convex regular polyhedron all lie on a sphere.
-All the dihedral angles of the polyhedron are equal.
-All the polyhedrons consisting of vertex figures are regular polygons.
-All the solid angles of the polyhedron are congruent.