**Faces**– All the flat surfaces of the three 3-D shapes are the faces. Solid shapes are made up of these plane figures called faces.**Edges**– The line segments which make the structure of the solid shapes are called edges. The two faces meet at the edges of the 3D shapes.**Vertex**– The corner of the solid shapes is called vertex. The two edges meet at the vertex. The plural of the vertex is vertices.

**Polyhedrons**

Polygons are the flat surface made up of line segments. The 3-D shapes made up of polygons are called polyhedron.

- These solid shapes have faces, edges and vertices.
- The polygons are the faces of the solid shape.
- Three or more edges meet at a point to form a vertex.
- The plural of word polyhedron is polyhedral.

**Non-polyhedron**

The solid shape who’s all the faces are not polygon are called non-polyhedron. i.e. it has one of the curved faces.

**Convex Polyhedrons**

If the line segment formed by joining any two vertices of the polyhedron lies inside the figure then it is said to be a convex polyhedron.

**Non-convex or Concave Polyhedron**

If anyone or more line segments formed by joining any two vertices of the polyhedron lie outside the figure then it is said to be a non-convex polyhedron.

**Regular Polyhedron**

If all the faces of a polyhedron are regular polygons and its same number of faces meets at each vertex then it is called regular polyhedron.

**Non-regular Polyhedron**

The polyhedron which is not regular is called non-regular polyhedron. Its vertices are not made by the same number of faces.

In this figure, 4 faces meet at the top point and 3 faces meet at all the bottom points.

**Prism**

If the top and bottom of a polyhedron are a congruent polygon and its lateral faces are parallelogram in shape, then it is said to be a prism.

**Pyramid**

If the base of a polyhedron is the polygon and its lateral faces are triangular in shape with a common vertex, then it is said to be a pyramid.

**Hollow hexagonal Prism**: Formed by joining two hexagons and six rectangles as shown below:

**Hollow Cylinder**: A cylinder is made by rotating a rectangle around either its length or breadth as shown below.

Solid | Number of Faces | Number of Edges | Number of Vertices |

Cube | 6 | 12 | 8 |

Rectangular Prism | 6 | 12 | 8 |

Triangular Prism | 5 | 9 | 6 |

Pentagonal Prism | 7 | 15 | 10 |

Hexagonal Prism | 8 | 18 | 12 |

Square Pyramid | 5 | 8 | 5 |

Triangular Pyramid | 4 | 6 | 6 |

Pentagonal Pyramid | 6 | 10 | 6 |

Hexagonal Pyramid | 7 | 12 | 7 |

**Euler’s formula**

Euler’s formula shows the relationship between edges, faces and vertices of a polyhedron.

Every polyhedron will satisfy the criterion F + V – E = 2,

Where F is the number of faces of the polyhedron, V is the vertices of the polyhedron and E is the number of edges of the polyhedron.

**Example**

Using Euler’s formula, find the number of faces if the number of vertices is 6 and the number of edges is 12.

**Solution**

Given, V = 6 and E = 12.

We know Euler’s formula, F + V – E = 2

So, F + 6 – 12 = 2.

Hence, F = 8.

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