Revision Notes on Three-Dimensional Figure

SOLID SHAPES

Shapes which have third dimension also along with 2-dimension of note book are called solid

shapes. We can feel solid shape or 3-Dimensional shape.

Two-dimensional shapes

Plane figures with only two measurements –length and width are called 2-D shapes.

Three-dimensional shapes

Solid figures with three measurements –length, width and height are called 3D shapes.

Cuboids: – A Solid shape that length, breadth and height are different is called cuboids.  

Ex.: – a match, a brick, a book, a geometrical box, a refrigerator, an almirah, etc.

Each plane surface of cuboids is called its face. (6-Faces)

Two adjacent faces of cuboids meet in an edge. (12-Edges)

Two edges meet at a corner called a vertex.        (8- Vertices)                                                   

All the faces of cuboids are not equal, only the opposite faces are equal.

All the edges of cuboids are not equal, only the opposite edges are equal.

Cube: – A Solid shape that length, breadth, and height are equal is called cube.

Ex.: – a dice, a chalk box, a ice-cube, a sugar-cubes etc. are called cube.

It’s all faces are of the same shape and size.

It’s all edges are of the same shape and size.

A cube has 6 rectangular faces, 12 edges and 8 vertices.

Prism

A solid whose two faces are parallel plane polygons and the side faces are rectangles is called prism.

Triangular Prism

The prism that is made up of two triangular faces and three rectangular faces is called triangular prism.

Two triangular faces- ABC & DEF

Three rectangular faces- ABED, BCEF & ACED

Pyramid: – A Solids shape with a flat base and three or four triangular sides is called pyramid.

Faces-4 Vertices-4 Edges-6

A pyramid has a polygonal base and flat triangular faces, which join at a common point called the apex.

A pyramid is formed by connecting the bases to an apex. Each edge of the base is connected to the apex, and forms the triangular face, called the lateral face.

If a pyramid has an n-sided base, then it has n+1 faces, n+1 vertices, and 2n edges.

 

Types Of Pyramids

Based on the shape of the base, the pyramid is classified into different types. Now, let us discuss the different types of pyramid shape one by one.

Triangular Pyramid

If the base of the pyramid is in a triangular shape (base with 3 sides), then the pyramid is called a triangular pyramid. As the triangle has 3 sides, then the triangular pyramid has the following properties:

No. of Faces = (3+1) = 4

No. of Vertices: (3+1) = 4

No. of Edges: 2(3) = 6

Square Pyramid

If the base of the pyramid is in the shape of a square (base with 4 sides), then it is called a square pyramid. As the square has 4 sides, then the square pyramid has the following properties:

No. of Faces: (4+1) = 5

No. of Vertices: (4+1)= 5

No. of Edges: 2(4) = 8

 

Number of faces, vertices and edges of some polyhedrons

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

Cone: – A solid triangular shape that base is circular is called cone.                                      

Ex: – Conical tents, Funnel, ice-cream cone, sharpened end of a pencil etc.

It has one plane face which is its base.        

It has only one curved face.

It has one vertex and one edge.

Cylinder: – A Solid long shape that bases are circular is called cylinder.

Ex: – Gas cylinders, Cold drink cans, Electric tube lights, circular pipes, or pillars etc.

Its top and base surface is plane surface. It is called bases of the cylinder.

Its curved face is called curved surface. It is called bases of the cylinder.

A cylinder has 2 round edges and no vertex.

It has 3 faces, 2 are plane and 1 is a curved face.

Sphere: – A Solids circular shape is called sphere.

It has no edge and no vertex.     It has only one curved face.

Ex: – Balls, Globes, apples, orange etc.                                                               

Comparisons of Properties of 3D shapes

As we already discussed above the properties of 3D shapes are based on their faces, edges and vertices. Thus, we can have a brief of all the properties here in the table.

Cube	6 square faces 8 vertices 12 edges
Cuboid	6 rectangular faces 8 vertices 12 edges
Cone	2 faces (circular base and curved surface) 1 vertex 1 edge
Cylinder	3 faces 2 edges 0 vertices
Sphere	1 curved surface 0 edges 0 vertices
Tetrahedron	4 faces 6 edges 4 vertices
Triangular prism	5 faces 9 edges 6 vertices
Square-based pyramid	5 faces 8 edges 5 vertices

Euler’s formula for 3-Dimensional Shapes

Euler’s formula shows the relationship between edges(E), faces(F) and vertices(V) of a polyhedron.

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

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.