Table of Contents

Last modified on March 28th, 2023

‘Poly’ – many, and ‘hedron’ – base or seat. So, a polyhedron is a 3-dimensional solid with multiple flat faces.

A polyhedron (plural – polyhedra or polyhedrons) is a 3-dimensional shape consisting of polygons joined at their edges. A polyhedron has no curved face. Its components are the multiple polygon-shaped flat faces.

It has 3 parts – face, edge, and vertex.

**Face**– The flat surface of a polyhedron.**Edge**– The region where 2 faces meet.**Vertex (Plural – vertices)**.- The point of intersection of 2 or more edges. It is also known as the corner of a polyhedron.

Polyhedrons are named based on the number of faces they have, such as Tetrahedron (4 faces), Pentahedron (5 faces), and Hexahedron (6 faces).

Platonic solids, prisms, and pyramids are 3 common groups of polyhedrons.

1. **Regular Polyhedron** – It is a polyhedron with faces that are regular polygons congruent to each other. Regular polyhedrons are also known as ‘platonic solids’. Cubes, tetrahedrons, and octahedrons are common examples of regular polyhedrons.

2. **Irregular Polyhedron **– It is a polyhedron with faces that are irregular polygons where all the components are not the same. Prisms and pyramids are common examples of irregular polyhedrons.

1. **Convex Polyhedrons** – It is a polyhedron where a line segment joining any 2 points on its surface lies completely inside it. All regular polyhedra are convex.

2. **Concave Polyhedron – **It is a polyhedron where a line segment joining any 2 points on its surface can lie outside it.

Like all other 3-dimensional shapes, we can calculate the surface areas and volumes of polyhedrons, such as a prism and a pyramid, using their specific formulas.

We can calculate the number of faces, edges, and vertices of any polyhedron using the formula based on Euler’s theorem:

**A 20-sided polyhedron has 30 edges. How many vertices does it have?**

Solution:

As we know that the given polyhedron is twenty sided, so it will have 20 faces,**V + F – E = 2**, here F = 20, E = 30

∴ V = 2 + E – F

= 2 + 30 – 20

= 12 vertices

**How many vertices does the polyhedron have if it is a 12-sided polyhedron with 30 edges?**

Solution:

As we know that the given polyhedron is a dodecahedron,**V + F – E = 2**, here F = 12, E = 30

∴ V = 2 + E – F

= 2 + 30 – 12

= 20 vertices

- Soccer balls
- The Great Pyramid of Giza
- Prisms used in Physics laboratory
- Polyhedron-shaped buildings
- Goldberg polyhedron

**Ans. **No. A cone is not a polyhedron because it has a curved face and a circular base.

**Ans. **No. A cylinder is not a polyhedron because it has a curved face and 2 circular bases.

**Ans. **Yes. A rectangular pyramid has 5 flat faces. So it is a pentahedron.

Last modified on March 28th, 2023